Theorem sltres 31815

Description: If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)

Assertion

Ref	Expression
sltres	|- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) -> A <s B ) )

Proof of Theorem sltres

Dummy variables a x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noreson 31813	. . . . . . 7 |- ( ( A e. No /\ X e. On ) -> ( A |` X ) e. No )
2	1	3adant2 1080	. . . . . 6 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( A |` X ) e. No )
3		noreson 31813	. . . . . . 7 |- ( ( B e. No /\ X e. On ) -> ( B |` X ) e. No )
4	3	3adant1 1079	. . . . . 6 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( B |` X ) e. No )
5		sltintdifex 31814	. . . . . . 7 |- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) <s ( B |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. _V ) )
6		onintrab 7001	. . . . . . 7 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. _V <-> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On )
7	5, 6	syl6ib 241	. . . . . 6 |- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) <s ( B |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) )
8	2, 4, 7	syl2anc 693	. . . . 5 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On ) )
9	8	imp 445	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On )
10		simpl3 1066	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> X e. On )
11		sltval2 31809	. . . . . . . . . . . 12 |- ( ( ( A |` X ) e. No /\ ( B |` X ) e. No ) -> ( ( A |` X ) <s ( B |` X ) <-> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) )
12	2, 4, 11	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) <-> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) )
13		fvex 6201	. . . . . . . . . . . . 13 |- ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V
14		fvex 6201	. . . . . . . . . . . . 13 |- ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V
15	13, 14	brtp 31639	. . . . . . . . . . . 12 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) )
16		1n0 7575	. . . . . . . . . . . . . . . . . 18 |- 1o =/= (/)
17	16	neii 2796	. . . . . . . . . . . . . . . . 17 |- -. 1o = (/)
18		eqeq1 2626	. . . . . . . . . . . . . . . . 17 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> 1o = (/) ) )
19	17, 18	mtbiri 317	. . . . . . . . . . . . . . . 16 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
20		ndmfv 6218	. . . . . . . . . . . . . . . 16 |- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
21	19, 20	nsyl2 142	. . . . . . . . . . . . . . 15 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
22	21	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
23	22	orcd 407	. . . . . . . . . . . . 13 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
24	21	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
25	24	orcd 407	. . . . . . . . . . . . 13 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
26		2on 7568	. . . . . . . . . . . . . . . . . . . . 21 |- 2o e. On
27	26	elexi 3213	. . . . . . . . . . . . . . . . . . . 20 |- 2o e. _V
28	27	prid2 4298	. . . . . . . . . . . . . . . . . . 19 |- 2o e. { 1o , 2o }
29	28	nosgnn0i 31812	. . . . . . . . . . . . . . . . . 18 |- (/) =/= 2o
30	29	neii 2796	. . . . . . . . . . . . . . . . 17 |- -. (/) = 2o
31		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> 2o = (/) ) )
32		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( 2o = (/) <-> (/) = 2o )
33	31, 32	syl6bb 276	. . . . . . . . . . . . . . . . 17 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) <-> (/) = 2o ) )
34	30, 33	mtbiri 317	. . . . . . . . . . . . . . . 16 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
35		ndmfv 6218	. . . . . . . . . . . . . . . 16 |- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
36	34, 35	nsyl2 142	. . . . . . . . . . . . . . 15 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) )
37	36	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) )
38	37	olcd 408	. . . . . . . . . . . . 13 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
39	23, 25, 38	3jaoi 1391	. . . . . . . . . . . 12 |- ( ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
40	15, 39	sylbi 207	. . . . . . . . . . 11 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
41	12, 40	syl6bi 243	. . . . . . . . . 10 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) ) )
42	41	imp 445	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
43		dmres 5419	. . . . . . . . . . . 12 |- dom ( A |` X ) = ( X i^i dom A )
44	43	elin2 3801	. . . . . . . . . . 11 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) <-> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) )
45	44	simplbi 476	. . . . . . . . . 10 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
46		dmres 5419	. . . . . . . . . . . 12 |- dom ( B |` X ) = ( X i^i dom B )
47	46	elin2 3801	. . . . . . . . . . 11 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) <-> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B ) )
48	47	simplbi 476	. . . . . . . . . 10 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
49	45, 48	jaoi 394	. . . . . . . . 9 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) \/ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
50	42, 49	syl 17	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
51		onelss 5766	. . . . . . . 8 |- ( X e. On -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ X ) )
52	10, 50, 51	sylc 65	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ X )
53	52	sselda 3603	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> y e. X )
54		onelon 5748	. . . . . . . . 9 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> y e. On )
55	9, 54	sylan 488	. . . . . . . 8 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> y e. On )
56		intss1 4492	. . . . . . . . . . . . 13 |- ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ y )
57		ontri1 5757	. . . . . . . . . . . . 13 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } C_ y <-> -. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
58	56, 57	syl5ib 234	. . . . . . . . . . . 12 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
59	58	con2d 129	. . . . . . . . . . 11 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ y e. On ) -> ( y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
60	9, 59	sylan 488	. . . . . . . . . 10 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. On ) -> ( y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
61	60	impancom 456	. . . . . . . . 9 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( y e. On -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
62	55, 61	mpd 15	. . . . . . . 8 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } )
63		fveq2 6191	. . . . . . . . . . . 12 |- ( a = y -> ( ( A |` X ) ` a ) = ( ( A |` X ) ` y ) )
64		fveq2 6191	. . . . . . . . . . . 12 |- ( a = y -> ( ( B |` X ) ` a ) = ( ( B |` X ) ` y ) )
65	63, 64	neeq12d 2855	. . . . . . . . . . 11 |- ( a = y -> ( ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) <-> ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) )
66	65	elrab 3363	. . . . . . . . . 10 |- ( y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } <-> ( y e. On /\ ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) )
67	66	simplbi2 655	. . . . . . . . 9 |- ( y e. On -> ( ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) -> y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
68	67	con3d 148	. . . . . . . 8 |- ( y e. On -> ( -. y e. { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) ) )
69	55, 62, 68	sylc 65	. . . . . . 7 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) )
70		df-ne 2795	. . . . . . . 8 |- ( ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) <-> -. ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) )
71	70	con2bii 347	. . . . . . 7 |- ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) <-> -. ( ( A |` X ) ` y ) =/= ( ( B |` X ) ` y ) )
72	69, 71	sylibr 224	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) )
73		fvres 6207	. . . . . . . 8 |- ( y e. X -> ( ( A |` X ) ` y ) = ( A ` y ) )
74		fvres 6207	. . . . . . . 8 |- ( y e. X -> ( ( B |` X ) ` y ) = ( B ` y ) )
75	73, 74	eqeq12d 2637	. . . . . . 7 |- ( y e. X -> ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) <-> ( A ` y ) = ( B ` y ) ) )
76	75	biimpd 219	. . . . . 6 |- ( y e. X -> ( ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) -> ( A ` y ) = ( B ` y ) ) )
77	53, 72, 76	sylc 65	. . . . 5 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) /\ y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( A ` y ) = ( B ` y ) )
78	77	ralrimiva 2966	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) )
79		fvresval 31665	. . . . . . . . . . . . . . 15 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) \/ ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
80	79	ori 390	. . . . . . . . . . . . . 14 |- ( -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
81	19, 80	nsyl2 142	. . . . . . . . . . . . 13 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
82	81	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
83		eqeq2 2633	. . . . . . . . . . . 12 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) )
84	82, 83	mpbid 222	. . . . . . . . . . 11 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o )
85	84	adantr 481	. . . . . . . . . 10 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o )
86	85	a1i 11	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o ) )
87	21	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
88	87, 45	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
89		nofun 31802	. . . . . . . . . . . . . . . . . 18 |- ( ( B |` X ) e. No -> Fun ( B |` X ) )
90		fvelrn 6352	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun ( B |` X ) /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) )
91	90	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( Fun ( B |` X ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) ) )
92	89, 91	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( B |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) ) )
93		norn 31804	. . . . . . . . . . . . . . . . . 18 |- ( ( B |` X ) e. No -> ran ( B |` X ) C_ { 1o , 2o } )
94	93	sseld 3602	. . . . . . . . . . . . . . . . 17 |- ( ( B |` X ) e. No -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) )
95	92, 94	syld 47	. . . . . . . . . . . . . . . 16 |- ( ( B |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) )
96		nosgnn0 31811	. . . . . . . . . . . . . . . . 17 |- -. (/) e. { 1o , 2o }
97		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) )
98	96, 97	mtbiri 317	. . . . . . . . . . . . . . . 16 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } )
99	95, 98	nsyli 155	. . . . . . . . . . . . . . 15 |- ( ( B |` X ) e. No -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
100	4, 99	syl 17	. . . . . . . . . . . . . 14 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
101	100	imp 445	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) )
102	101	adantrl 752	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) )
103	47	simplbi2 655	. . . . . . . . . . . . 13 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) ) )
104	103	con3d 148	. . . . . . . . . . . 12 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B ) )
105	88, 102, 104	sylc 65	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B )
106		ndmfv 6218	. . . . . . . . . . 11 |- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom B -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
107	105, 106	syl 17	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
108	107	ex 450	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) )
109	86, 108	jcad 555	. . . . . . . 8 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) ) )
110		fvresval 31665	. . . . . . . . . . . . . 14 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) \/ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
111	110	ori 390	. . . . . . . . . . . . 13 |- ( -. ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
112	34, 111	nsyl2 142	. . . . . . . . . . . 12 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
113	112	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
114		eqeq2 2633	. . . . . . . . . . 11 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) )
115	113, 114	mpbid 222	. . . . . . . . . 10 |- ( ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o )
116	84, 115	anim12i 590	. . . . . . . . 9 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) )
117	116	a1i 11	. . . . . . . 8 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) )
118	36	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( B |` X ) )
119	118, 48	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X )
120		nofun 31802	. . . . . . . . . . . . . . . . . 18 |- ( ( A |` X ) e. No -> Fun ( A |` X ) )
121		fvelrn 6352	. . . . . . . . . . . . . . . . . . 19 |- ( ( Fun ( A |` X ) /\ |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) )
122	121	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( Fun ( A |` X ) -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) ) )
123	120, 122	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( A |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) ) )
124		norn 31804	. . . . . . . . . . . . . . . . . 18 |- ( ( A |` X ) e. No -> ran ( A |` X ) C_ { 1o , 2o } )
125	124	sseld 3602	. . . . . . . . . . . . . . . . 17 |- ( ( A |` X ) e. No -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. ran ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) )
126	123, 125	syld 47	. . . . . . . . . . . . . . . 16 |- ( ( A |` X ) e. No -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } ) )
127		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) )
128	96, 127	mtbiri 317	. . . . . . . . . . . . . . . 16 |- ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. { 1o , 2o } )
129	126, 128	nsyli 155	. . . . . . . . . . . . . . 15 |- ( ( A |` X ) e. No -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) )
130	2, 129	syl 17	. . . . . . . . . . . . . 14 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) )
131	130	imp 445	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
132	131	adantrr 753	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) )
133	44	simplbi2 655	. . . . . . . . . . . . 13 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A -> |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) ) )
134	133	con3d 148	. . . . . . . . . . . 12 |- ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. X -> ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom ( A |` X ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) )
135	119, 132, 134	sylc 65	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A )
136	135	ex 450	. . . . . . . . . 10 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A ) )
137		ndmfv 6218	. . . . . . . . . 10 |- ( -. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. dom A -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) )
138	136, 137	syl6 35	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) )
139	115	adantl 482	. . . . . . . . . 10 |- ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o )
140	139	a1i 11	. . . . . . . . 9 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) )
141	138, 140	jcad 555	. . . . . . . 8 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) -> ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) )
142	109, 117, 141	3orim123d 1407	. . . . . . 7 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) -> ( ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) ) )
143		fvex 6201	. . . . . . . 8 |- ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V
144		fvex 6201	. . . . . . . 8 |- ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) e. _V
145	143, 144	brtp 31639	. . . . . . 7 |- ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) <-> ( ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 1o /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) \/ ( ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = (/) /\ ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) = 2o ) ) )
146	142, 15, 145	3imtr4g 285	. . . . . 6 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( ( A |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( ( B |` X ) ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) )
147	12, 146	sylbid 230	. . . . 5 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) )
148	147	imp 445	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
149		raleq 3138	. . . . . 6 |- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( A. y e. x ( A ` y ) = ( B ` y ) <-> A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) ) )
150		fveq2 6191	. . . . . . 7 |- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( A ` x ) = ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
151		fveq2 6191	. . . . . . 7 |- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( B ` x ) = ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) )
152	150, 151	breq12d 4666	. . . . . 6 |- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) <-> ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) )
153	149, 152	anbi12d 747	. . . . 5 |- ( x = |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } -> ( ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) <-> ( A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) )
154	153	rspcev 3309	. . . 4 |- ( ( |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } e. On /\ ( A. y e. |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ( A ` y ) = ( B ` y ) /\ ( A ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` |^| { a e. On | ( ( A |` X ) ` a ) =/= ( ( B |` X ) ` a ) } ) ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
155	9, 78, 148, 154	syl12anc 1324	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
156		sltval 31800	. . . . 5 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
157	156	3adant3 1081	. . . 4 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
158	157	adantr 481	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
159	155, 158	mpbird 247	. 2 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( A |` X ) <s ( B |` X ) ) -> A <s B )
160	159	ex 450	1 |- ( ( A e. No /\ B e. No /\ X e. On ) -> ( ( A |` X ) <s ( B |` X ) -> A <s B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {cpr 4179   {ctp 4181   <.cop 4183   |^|cint 4475   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Oncon0 5723   Fun wfun 5882   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797

This theorem is referenced by:  noresle  31846  nosupbnd1lem1  31854  nosupbnd1lem2  31855  nosupbnd1  31860  nosupbnd2lem1  31861  nosupbnd2  31862  noetalem2  31864  noetalem3  31865