Theorem f1oun2prg 13662

Description: A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Assertion

Ref	Expression
f1oun2prg	|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) : ( { 0 , 1 } u. { 2 , 3 } ) -1-1-onto-> ( { A , B } u. { C , D } ) ) )

Proof of Theorem f1oun2prg

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> A e. V )
2		0z 11388	. . . . . . 7 |- 0 e. ZZ
3	1, 2	jctil 560	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( 0 e. ZZ /\ A e. V ) )
4	3	ad2antrr 762	. . . . 5 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( 0 e. ZZ /\ A e. V ) )
5		simpr 477	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> B e. W )
6		1z 11407	. . . . . . 7 |- 1 e. ZZ
7	5, 6	jctil 560	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( 1 e. ZZ /\ B e. W ) )
8	7	ad2antrr 762	. . . . 5 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( 1 e. ZZ /\ B e. W ) )
9	4, 8	jca 554	. . . 4 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( 0 e. ZZ /\ A e. V ) /\ ( 1 e. ZZ /\ B e. W ) ) )
10		id 22	. . . . . . . 8 |- ( A =/= B -> A =/= B )
11	10	3ad2ant1 1082	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ A =/= D ) -> A =/= B )
12		0ne1 11088	. . . . . . 7 |- 0 =/= 1
13	11, 12	jctil 560	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ A =/= D ) -> ( 0 =/= 1 /\ A =/= B ) )
14	13	adantr 481	. . . . 5 |- ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( 0 =/= 1 /\ A =/= B ) )
15	14	adantl 482	. . . 4 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( 0 =/= 1 /\ A =/= B ) )
16		f1oprg 6181	. . . 4 |- ( ( ( 0 e. ZZ /\ A e. V ) /\ ( 1 e. ZZ /\ B e. W ) ) -> ( ( 0 =/= 1 /\ A =/= B ) -> { <. 0 , A >. , <. 1 , B >. } : { 0 , 1 } -1-1-onto-> { A , B } ) )
17	9, 15, 16	sylc 65	. . 3 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> { <. 0 , A >. , <. 1 , B >. } : { 0 , 1 } -1-1-onto-> { A , B } )
18		simpl 473	. . . . . . . 8 |- ( ( C e. X /\ D e. Y ) -> C e. X )
19		2nn 11185	. . . . . . . 8 |- 2 e. NN
20	18, 19	jctil 560	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> ( 2 e. NN /\ C e. X ) )
21	20	adantl 482	. . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( 2 e. NN /\ C e. X ) )
22		simpr 477	. . . . . . . 8 |- ( ( C e. X /\ D e. Y ) -> D e. Y )
23		3nn 11186	. . . . . . . 8 |- 3 e. NN
24	22, 23	jctil 560	. . . . . . 7 |- ( ( C e. X /\ D e. Y ) -> ( 3 e. NN /\ D e. Y ) )
25	24	adantl 482	. . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( 3 e. NN /\ D e. Y ) )
26	21, 25	jca 554	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( 2 e. NN /\ C e. X ) /\ ( 3 e. NN /\ D e. Y ) ) )
27	26	adantr 481	. . . 4 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( 2 e. NN /\ C e. X ) /\ ( 3 e. NN /\ D e. Y ) ) )
28		id 22	. . . . . . . 8 |- ( C =/= D -> C =/= D )
29	28	3ad2ant3 1084	. . . . . . 7 |- ( ( B =/= C /\ B =/= D /\ C =/= D ) -> C =/= D )
30		2re 11090	. . . . . . . 8 |- 2 e. RR
31		2lt3 11195	. . . . . . . 8 |- 2 < 3
32	30, 31	ltneii 10150	. . . . . . 7 |- 2 =/= 3
33	29, 32	jctil 560	. . . . . 6 |- ( ( B =/= C /\ B =/= D /\ C =/= D ) -> ( 2 =/= 3 /\ C =/= D ) )
34	33	adantl 482	. . . . 5 |- ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( 2 =/= 3 /\ C =/= D ) )
35	34	adantl 482	. . . 4 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( 2 =/= 3 /\ C =/= D ) )
36		f1oprg 6181	. . . 4 |- ( ( ( 2 e. NN /\ C e. X ) /\ ( 3 e. NN /\ D e. Y ) ) -> ( ( 2 =/= 3 /\ C =/= D ) -> { <. 2 , C >. , <. 3 , D >. } : { 2 , 3 } -1-1-onto-> { C , D } ) )
37	27, 35, 36	sylc 65	. . 3 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> { <. 2 , C >. , <. 3 , D >. } : { 2 , 3 } -1-1-onto-> { C , D } )
38		disjsn2 4247	. . . . . . . . . 10 |- ( A =/= C -> ( { A } i^i { C } ) = (/) )
39	38	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A =/= B /\ A =/= C /\ A =/= D ) -> ( { A } i^i { C } ) = (/) )
40		disjsn2 4247	. . . . . . . . . 10 |- ( B =/= C -> ( { B } i^i { C } ) = (/) )
41	40	3ad2ant1 1082	. . . . . . . . 9 |- ( ( B =/= C /\ B =/= D /\ C =/= D ) -> ( { B } i^i { C } ) = (/) )
42	39, 41	anim12i 590	. . . . . . . 8 |- ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
43	42	adantl 482	. . . . . . 7 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
44		df-pr 4180	. . . . . . . . . 10 |- { A , B } = ( { A } u. { B } )
45	44	ineq1i 3810	. . . . . . . . 9 |- ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } )
46	45	eqeq1i 2627	. . . . . . . 8 |- ( ( { A , B } i^i { C } ) = (/) <-> ( ( { A } u. { B } ) i^i { C } ) = (/) )
47		undisj1 4029	. . . . . . . 8 |- ( ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) <-> ( ( { A } u. { B } ) i^i { C } ) = (/) )
48	46, 47	bitr4i 267	. . . . . . 7 |- ( ( { A , B } i^i { C } ) = (/) <-> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
49	43, 48	sylibr 224	. . . . . 6 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( { A , B } i^i { C } ) = (/) )
50		disjsn2 4247	. . . . . . . . . 10 |- ( A =/= D -> ( { A } i^i { D } ) = (/) )
51	50	3ad2ant3 1084	. . . . . . . . 9 |- ( ( A =/= B /\ A =/= C /\ A =/= D ) -> ( { A } i^i { D } ) = (/) )
52		disjsn2 4247	. . . . . . . . . 10 |- ( B =/= D -> ( { B } i^i { D } ) = (/) )
53	52	3ad2ant2 1083	. . . . . . . . 9 |- ( ( B =/= C /\ B =/= D /\ C =/= D ) -> ( { B } i^i { D } ) = (/) )
54	51, 53	anim12i 590	. . . . . . . 8 |- ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
55	54	adantl 482	. . . . . . 7 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
56	44	ineq1i 3810	. . . . . . . . 9 |- ( { A , B } i^i { D } ) = ( ( { A } u. { B } ) i^i { D } )
57	56	eqeq1i 2627	. . . . . . . 8 |- ( ( { A , B } i^i { D } ) = (/) <-> ( ( { A } u. { B } ) i^i { D } ) = (/) )
58		undisj1 4029	. . . . . . . 8 |- ( ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) <-> ( ( { A } u. { B } ) i^i { D } ) = (/) )
59	57, 58	bitr4i 267	. . . . . . 7 |- ( ( { A , B } i^i { D } ) = (/) <-> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
60	55, 59	sylibr 224	. . . . . 6 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( { A , B } i^i { D } ) = (/) )
61	49, 60	jca 554	. . . . 5 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( { A , B } i^i { C } ) = (/) /\ ( { A , B } i^i { D } ) = (/) ) )
62		undisj2 4030	. . . . . 6 |- ( ( ( { A , B } i^i { C } ) = (/) /\ ( { A , B } i^i { D } ) = (/) ) <-> ( { A , B } i^i ( { C } u. { D } ) ) = (/) )
63		df-pr 4180	. . . . . . . . 9 |- { C , D } = ( { C } u. { D } )
64	63	eqcomi 2631	. . . . . . . 8 |- ( { C } u. { D } ) = { C , D }
65	64	ineq2i 3811	. . . . . . 7 |- ( { A , B } i^i ( { C } u. { D } ) ) = ( { A , B } i^i { C , D } )
66	65	eqeq1i 2627	. . . . . 6 |- ( ( { A , B } i^i ( { C } u. { D } ) ) = (/) <-> ( { A , B } i^i { C , D } ) = (/) )
67	62, 66	bitri 264	. . . . 5 |- ( ( ( { A , B } i^i { C } ) = (/) /\ ( { A , B } i^i { D } ) = (/) ) <-> ( { A , B } i^i { C , D } ) = (/) )
68	61, 67	sylib 208	. . . 4 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( { A , B } i^i { C , D } ) = (/) )
69		df-pr 4180	. . . . . . . . 9 |- { 0 , 1 } = ( { 0 } u. { 1 } )
70	69	eqcomi 2631	. . . . . . . 8 |- ( { 0 } u. { 1 } ) = { 0 , 1 }
71	70	ineq1i 3810	. . . . . . 7 |- ( ( { 0 } u. { 1 } ) i^i { 2 } ) = ( { 0 , 1 } i^i { 2 } )
72		0ne2 11239	. . . . . . . . . 10 |- 0 =/= 2
73		disjsn2 4247	. . . . . . . . . 10 |- ( 0 =/= 2 -> ( { 0 } i^i { 2 } ) = (/) )
74	72, 73	ax-mp 5	. . . . . . . . 9 |- ( { 0 } i^i { 2 } ) = (/)
75		1ne2 11240	. . . . . . . . . 10 |- 1 =/= 2
76		disjsn2 4247	. . . . . . . . . 10 |- ( 1 =/= 2 -> ( { 1 } i^i { 2 } ) = (/) )
77	75, 76	ax-mp 5	. . . . . . . . 9 |- ( { 1 } i^i { 2 } ) = (/)
78	74, 77	pm3.2i 471	. . . . . . . 8 |- ( ( { 0 } i^i { 2 } ) = (/) /\ ( { 1 } i^i { 2 } ) = (/) )
79		undisj1 4029	. . . . . . . 8 |- ( ( ( { 0 } i^i { 2 } ) = (/) /\ ( { 1 } i^i { 2 } ) = (/) ) <-> ( ( { 0 } u. { 1 } ) i^i { 2 } ) = (/) )
80	78, 79	mpbi 220	. . . . . . 7 |- ( ( { 0 } u. { 1 } ) i^i { 2 } ) = (/)
81	71, 80	eqtr3i 2646	. . . . . 6 |- ( { 0 , 1 } i^i { 2 } ) = (/)
82	70	ineq1i 3810	. . . . . . 7 |- ( ( { 0 } u. { 1 } ) i^i { 3 } ) = ( { 0 , 1 } i^i { 3 } )
83		3ne0 11115	. . . . . . . . . . 11 |- 3 =/= 0
84	83	necomi 2848	. . . . . . . . . 10 |- 0 =/= 3
85		disjsn2 4247	. . . . . . . . . 10 |- ( 0 =/= 3 -> ( { 0 } i^i { 3 } ) = (/) )
86	84, 85	ax-mp 5	. . . . . . . . 9 |- ( { 0 } i^i { 3 } ) = (/)
87		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
88		1lt3 11196	. . . . . . . . . . 11 |- 1 < 3
89	87, 88	ltneii 10150	. . . . . . . . . 10 |- 1 =/= 3
90		disjsn2 4247	. . . . . . . . . 10 |- ( 1 =/= 3 -> ( { 1 } i^i { 3 } ) = (/) )
91	89, 90	ax-mp 5	. . . . . . . . 9 |- ( { 1 } i^i { 3 } ) = (/)
92	86, 91	pm3.2i 471	. . . . . . . 8 |- ( ( { 0 } i^i { 3 } ) = (/) /\ ( { 1 } i^i { 3 } ) = (/) )
93		undisj1 4029	. . . . . . . 8 |- ( ( ( { 0 } i^i { 3 } ) = (/) /\ ( { 1 } i^i { 3 } ) = (/) ) <-> ( ( { 0 } u. { 1 } ) i^i { 3 } ) = (/) )
94	92, 93	mpbi 220	. . . . . . 7 |- ( ( { 0 } u. { 1 } ) i^i { 3 } ) = (/)
95	82, 94	eqtr3i 2646	. . . . . 6 |- ( { 0 , 1 } i^i { 3 } ) = (/)
96	81, 95	pm3.2i 471	. . . . 5 |- ( ( { 0 , 1 } i^i { 2 } ) = (/) /\ ( { 0 , 1 } i^i { 3 } ) = (/) )
97		undisj2 4030	. . . . . 6 |- ( ( ( { 0 , 1 } i^i { 2 } ) = (/) /\ ( { 0 , 1 } i^i { 3 } ) = (/) ) <-> ( { 0 , 1 } i^i ( { 2 } u. { 3 } ) ) = (/) )
98		df-pr 4180	. . . . . . . . 9 |- { 2 , 3 } = ( { 2 } u. { 3 } )
99	98	eqcomi 2631	. . . . . . . 8 |- ( { 2 } u. { 3 } ) = { 2 , 3 }
100	99	ineq2i 3811	. . . . . . 7 |- ( { 0 , 1 } i^i ( { 2 } u. { 3 } ) ) = ( { 0 , 1 } i^i { 2 , 3 } )
101	100	eqeq1i 2627	. . . . . 6 |- ( ( { 0 , 1 } i^i ( { 2 } u. { 3 } ) ) = (/) <-> ( { 0 , 1 } i^i { 2 , 3 } ) = (/) )
102	97, 101	bitri 264	. . . . 5 |- ( ( ( { 0 , 1 } i^i { 2 } ) = (/) /\ ( { 0 , 1 } i^i { 3 } ) = (/) ) <-> ( { 0 , 1 } i^i { 2 , 3 } ) = (/) )
103	96, 102	mpbi 220	. . . 4 |- ( { 0 , 1 } i^i { 2 , 3 } ) = (/)
104	68, 103	jctil 560	. . 3 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( ( { 0 , 1 } i^i { 2 , 3 } ) = (/) /\ ( { A , B } i^i { C , D } ) = (/) ) )
105		f1oun 6156	. . 3 |- ( ( ( { <. 0 , A >. , <. 1 , B >. } : { 0 , 1 } -1-1-onto-> { A , B } /\ { <. 2 , C >. , <. 3 , D >. } : { 2 , 3 } -1-1-onto-> { C , D } ) /\ ( ( { 0 , 1 } i^i { 2 , 3 } ) = (/) /\ ( { A , B } i^i { C , D } ) = (/) ) ) -> ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) : ( { 0 , 1 } u. { 2 , 3 } ) -1-1-onto-> ( { A , B } u. { C , D } ) )
106	17, 37, 104, 105	syl21anc 1325	. 2 |- ( ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) /\ ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) ) -> ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) : ( { 0 , 1 } u. { 2 , 3 } ) -1-1-onto-> ( { A , B } u. { C , D } ) )
107	106	ex 450	1 |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= B /\ A =/= C /\ A =/= D ) /\ ( B =/= C /\ B =/= D /\ C =/= D ) ) -> ( { <. 0 , A >. , <. 1 , B >. } u. { <. 2 , C >. , <. 3 , D >. } ) : ( { 0 , 1 } u. { 2 , 3 } ) -1-1-onto-> ( { A , B } u. { C , D } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   -1-1-onto->wf1o 5887   0cc0 9936   1c1 9937   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-z 11378

This theorem is referenced by:  s4f1o  13663