Theorem nolesgn2ores 31825

Description: Given A less than or equal to B, equal to B up to X, and A ( X ) = 2o, then ( A |` suc X ) = ( B |` suc X ). (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolesgn2ores	|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( A |` suc X ) = ( B |` suc X ) )

Proof of Theorem nolesgn2ores

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmres 5419	. . . 4 |- dom ( A |` suc X ) = ( suc X i^i dom A )
2		simp11 1091	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> A e. No )
3		nodmord 31806	. . . . . . 7 |- ( A e. No -> Ord dom A )
4	2, 3	syl 17	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> Ord dom A )
5		ndmfv 6218	. . . . . . . . . 10 |- ( -. X e. dom A -> ( A ` X ) = (/) )
6		2on 7568	. . . . . . . . . . . . . . 15 |- 2o e. On
7	6	elexi 3213	. . . . . . . . . . . . . 14 |- 2o e. _V
8	7	prid2 4298	. . . . . . . . . . . . 13 |- 2o e. { 1o , 2o }
9	8	nosgnn0i 31812	. . . . . . . . . . . 12 |- (/) =/= 2o
10		neeq1 2856	. . . . . . . . . . . 12 |- ( ( A ` X ) = (/) -> ( ( A ` X ) =/= 2o <-> (/) =/= 2o ) )
11	9, 10	mpbiri 248	. . . . . . . . . . 11 |- ( ( A ` X ) = (/) -> ( A ` X ) =/= 2o )
12	11	neneqd 2799	. . . . . . . . . 10 |- ( ( A ` X ) = (/) -> -. ( A ` X ) = 2o )
13	5, 12	syl 17	. . . . . . . . 9 |- ( -. X e. dom A -> -. ( A ` X ) = 2o )
14	13	con4i 113	. . . . . . . 8 |- ( ( A ` X ) = 2o -> X e. dom A )
15	14	adantl 482	. . . . . . 7 |- ( ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) -> X e. dom A )
16	15	3ad2ant2 1083	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> X e. dom A )
17		ordsucss 7018	. . . . . 6 |- ( Ord dom A -> ( X e. dom A -> suc X C_ dom A ) )
18	4, 16, 17	sylc 65	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> suc X C_ dom A )
19		df-ss 3588	. . . . 5 |- ( suc X C_ dom A <-> ( suc X i^i dom A ) = suc X )
20	18, 19	sylib 208	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( suc X i^i dom A ) = suc X )
21	1, 20	syl5eq 2668	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> dom ( A |` suc X ) = suc X )
22		dmres 5419	. . . 4 |- dom ( B |` suc X ) = ( suc X i^i dom B )
23		simp12 1092	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> B e. No )
24		nodmord 31806	. . . . . . 7 |- ( B e. No -> Ord dom B )
25	23, 24	syl 17	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> Ord dom B )
26		nolesgn2o 31824	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( B ` X ) = 2o )
27		ndmfv 6218	. . . . . . . . 9 |- ( -. X e. dom B -> ( B ` X ) = (/) )
28		neeq1 2856	. . . . . . . . . . 11 |- ( ( B ` X ) = (/) -> ( ( B ` X ) =/= 2o <-> (/) =/= 2o ) )
29	9, 28	mpbiri 248	. . . . . . . . . 10 |- ( ( B ` X ) = (/) -> ( B ` X ) =/= 2o )
30	29	neneqd 2799	. . . . . . . . 9 |- ( ( B ` X ) = (/) -> -. ( B ` X ) = 2o )
31	27, 30	syl 17	. . . . . . . 8 |- ( -. X e. dom B -> -. ( B ` X ) = 2o )
32	31	con4i 113	. . . . . . 7 |- ( ( B ` X ) = 2o -> X e. dom B )
33	26, 32	syl 17	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> X e. dom B )
34		ordsucss 7018	. . . . . 6 |- ( Ord dom B -> ( X e. dom B -> suc X C_ dom B ) )
35	25, 33, 34	sylc 65	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> suc X C_ dom B )
36		df-ss 3588	. . . . 5 |- ( suc X C_ dom B <-> ( suc X i^i dom B ) = suc X )
37	35, 36	sylib 208	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( suc X i^i dom B ) = suc X )
38	22, 37	syl5eq 2668	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> dom ( B |` suc X ) = suc X )
39	21, 38	eqtr4d 2659	. 2 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> dom ( A |` suc X ) = dom ( B |` suc X ) )
40	21	eleq2d 2687	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x e. dom ( A |` suc X ) <-> x e. suc X ) )
41		vex 3203	. . . . . . . . 9 |- x e. _V
42	41	elsuc 5794	. . . . . . . 8 |- ( x e. suc X <-> ( x e. X \/ x = X ) )
43		simp2l 1087	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( A |` X ) = ( B |` X ) )
44	43	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( ( A |` X ) ` x ) = ( ( B |` X ) ` x ) )
45	44	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. X ) -> ( ( A |` X ) ` x ) = ( ( B |` X ) ` x ) )
46		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. X ) -> x e. X )
47	46	fvresd 6208	. . . . . . . . . . 11 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. X ) -> ( ( A |` X ) ` x ) = ( A ` x ) )
48	46	fvresd 6208	. . . . . . . . . . 11 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. X ) -> ( ( B |` X ) ` x ) = ( B ` x ) )
49	45, 47, 48	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. X ) -> ( A ` x ) = ( B ` x ) )
50	49	ex 450	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x e. X -> ( A ` x ) = ( B ` x ) ) )
51		simp2r 1088	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( A ` X ) = 2o )
52	51, 26	eqtr4d 2659	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( A ` X ) = ( B ` X ) )
53		fveq2 6191	. . . . . . . . . . 11 |- ( x = X -> ( A ` x ) = ( A ` X ) )
54		fveq2 6191	. . . . . . . . . . 11 |- ( x = X -> ( B ` x ) = ( B ` X ) )
55	53, 54	eqeq12d 2637	. . . . . . . . . 10 |- ( x = X -> ( ( A ` x ) = ( B ` x ) <-> ( A ` X ) = ( B ` X ) ) )
56	52, 55	syl5ibrcom 237	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x = X -> ( A ` x ) = ( B ` x ) ) )
57	50, 56	jaod 395	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( ( x e. X \/ x = X ) -> ( A ` x ) = ( B ` x ) ) )
58	42, 57	syl5bi 232	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x e. suc X -> ( A ` x ) = ( B ` x ) ) )
59	58	imp 445	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. suc X ) -> ( A ` x ) = ( B ` x ) )
60		simpr 477	. . . . . . 7 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. suc X ) -> x e. suc X )
61	60	fvresd 6208	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. suc X ) -> ( ( A |` suc X ) ` x ) = ( A ` x ) )
62	60	fvresd 6208	. . . . . 6 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. suc X ) -> ( ( B |` suc X ) ` x ) = ( B ` x ) )
63	59, 61, 62	3eqtr4d 2666	. . . . 5 |- ( ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) /\ x e. suc X ) -> ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) )
64	63	ex 450	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x e. suc X -> ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) )
65	40, 64	sylbid 230	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( x e. dom ( A |` suc X ) -> ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) )
66	65	ralrimiv 2965	. 2 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) )
67		nofun 31802	. . . 4 |- ( A e. No -> Fun A )
68		funres 5929	. . . 4 |- ( Fun A -> Fun ( A |` suc X ) )
69	2, 67, 68	3syl 18	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> Fun ( A |` suc X ) )
70		nofun 31802	. . . 4 |- ( B e. No -> Fun B )
71		funres 5929	. . . 4 |- ( Fun B -> Fun ( B |` suc X ) )
72	23, 70, 71	3syl 18	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> Fun ( B |` suc X ) )
73		eqfunfv 6316	. . 3 |- ( ( Fun ( A |` suc X ) /\ Fun ( B |` suc X ) ) -> ( ( A |` suc X ) = ( B |` suc X ) <-> ( dom ( A |` suc X ) = dom ( B |` suc X ) /\ A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) )
74	69, 72, 73	syl2anc 693	. 2 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( ( A |` suc X ) = ( B |` suc X ) <-> ( dom ( A |` suc X ) = dom ( B |` suc X ) /\ A. x e. dom ( A |` suc X ) ( ( A |` suc X ) ` x ) = ( ( B |` suc X ) ` x ) ) ) )
75	39, 66, 74	mpbir2and 957	1 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( A |` suc X ) = ( B |` suc X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   |` cres 5116   Ord word 5722   Oncon0 5723   suc csuc 5725   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-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:  nosupbnd1lem3  31856