Theorem wfrlem5 7419

Description: Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfrlem5.1	|- R We A
wfrlem5.2	|- R Se A
wfrlem5.3	|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }

Assertion

Ref	Expression
wfrlem5	|- ( ( g e. B /\ h e. B ) -> ( ( x g u /\ x h v ) -> u = v ) )

Distinct variable groups:   A, f, g, h, x, y   f, F, g, h, x, y   R, f, g, h, x, y   u, g, v, h, x

Allowed substitution hints:   A( v, u)   B( x, y, v, u, f, g, h)   R( v, u)   F( v, u)

Proof of Theorem wfrlem5

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2		vex 3203	. . . . . 6 |- u e. _V
3	1, 2	breldm 5329	. . . . 5 |- ( x g u -> x e. dom g )
4		vex 3203	. . . . . 6 |- v e. _V
5	1, 4	breldm 5329	. . . . 5 |- ( x h v -> x e. dom h )
6	3, 5	anim12i 590	. . . 4 |- ( ( x g u /\ x h v ) -> ( x e. dom g /\ x e. dom h ) )
7		elin 3796	. . . 4 |- ( x e. ( dom g i^i dom h ) <-> ( x e. dom g /\ x e. dom h ) )
8	6, 7	sylibr 224	. . 3 |- ( ( x g u /\ x h v ) -> x e. ( dom g i^i dom h ) )
9		anandir 872	. . . 4 |- ( ( ( x g u /\ x h v ) /\ x e. ( dom g i^i dom h ) ) <-> ( ( x g u /\ x e. ( dom g i^i dom h ) ) /\ ( x h v /\ x e. ( dom g i^i dom h ) ) ) )
10	2	brres 5402	. . . . 5 |- ( x ( g |` ( dom g i^i dom h ) ) u <-> ( x g u /\ x e. ( dom g i^i dom h ) ) )
11	4	brres 5402	. . . . 5 |- ( x ( h |` ( dom g i^i dom h ) ) v <-> ( x h v /\ x e. ( dom g i^i dom h ) ) )
12	10, 11	anbi12i 733	. . . 4 |- ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) <-> ( ( x g u /\ x e. ( dom g i^i dom h ) ) /\ ( x h v /\ x e. ( dom g i^i dom h ) ) ) )
13	9, 12	sylbb2 228	. . 3 |- ( ( ( x g u /\ x h v ) /\ x e. ( dom g i^i dom h ) ) -> ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) )
14	8, 13	mpdan 702	. 2 |- ( ( x g u /\ x h v ) -> ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) )
15		wfrlem5.3	. . . . . . . . 9 |- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
16	15	wfrlem3 7416	. . . . . . . 8 |- ( g e. B -> dom g C_ A )
17		ssinss1 3841	. . . . . . . 8 |- ( dom g C_ A -> ( dom g i^i dom h ) C_ A )
18		wfrlem5.1	. . . . . . . . . 10 |- R We A
19		wess 5101	. . . . . . . . . 10 |- ( ( dom g i^i dom h ) C_ A -> ( R We A -> R We ( dom g i^i dom h ) ) )
20	18, 19	mpi 20	. . . . . . . . 9 |- ( ( dom g i^i dom h ) C_ A -> R We ( dom g i^i dom h ) )
21		wfrlem5.2	. . . . . . . . . 10 |- R Se A
22		sess2 5083	. . . . . . . . . 10 |- ( ( dom g i^i dom h ) C_ A -> ( R Se A -> R Se ( dom g i^i dom h ) ) )
23	21, 22	mpi 20	. . . . . . . . 9 |- ( ( dom g i^i dom h ) C_ A -> R Se ( dom g i^i dom h ) )
24	20, 23	jca 554	. . . . . . . 8 |- ( ( dom g i^i dom h ) C_ A -> ( R We ( dom g i^i dom h ) /\ R Se ( dom g i^i dom h ) ) )
25	16, 17, 24	3syl 18	. . . . . . 7 |- ( g e. B -> ( R We ( dom g i^i dom h ) /\ R Se ( dom g i^i dom h ) ) )
26	25	adantr 481	. . . . . 6 |- ( ( g e. B /\ h e. B ) -> ( R We ( dom g i^i dom h ) /\ R Se ( dom g i^i dom h ) ) )
27	18, 15	wfrlem4 7418	. . . . . 6 |- ( ( g e. B /\ h e. B ) -> ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) )
28	18, 15	wfrlem4 7418	. . . . . . . 8 |- ( ( h e. B /\ g e. B ) -> ( ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) /\ A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) )
29	28	ancoms 469	. . . . . . 7 |- ( ( g e. B /\ h e. B ) -> ( ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) /\ A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) )
30		incom 3805	. . . . . . . . . . 11 |- ( dom g i^i dom h ) = ( dom h i^i dom g )
31	30	reseq2i 5393	. . . . . . . . . 10 |- ( h |` ( dom g i^i dom h ) ) = ( h |` ( dom h i^i dom g ) )
32	31	fneq1i 5985	. . . . . . . . 9 |- ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) <-> ( h |` ( dom h i^i dom g ) ) Fn ( dom g i^i dom h ) )
33	30	fneq2i 5986	. . . . . . . . 9 |- ( ( h |` ( dom h i^i dom g ) ) Fn ( dom g i^i dom h ) <-> ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) )
34	32, 33	bitri 264	. . . . . . . 8 |- ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) <-> ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) )
35	31	fveq1i 6192	. . . . . . . . . 10 |- ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( ( h |` ( dom h i^i dom g ) ) ` a )
36		predeq2 5683	. . . . . . . . . . . . 13 |- ( ( dom g i^i dom h ) = ( dom h i^i dom g ) -> Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , ( dom h i^i dom g ) , a ) )
37	30, 36	ax-mp 5	. . . . . . . . . . . 12 |- Pred ( R , ( dom g i^i dom h ) , a ) = Pred ( R , ( dom h i^i dom g ) , a )
38	31, 37	reseq12i 5394	. . . . . . . . . . 11 |- ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) = ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) )
39	38	fveq2i 6194	. . . . . . . . . 10 |- ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) )
40	35, 39	eqeq12i 2636	. . . . . . . . 9 |- ( ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) <-> ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) )
41	30, 40	raleqbii 2990	. . . . . . . 8 |- ( A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) <-> A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) )
42	34, 41	anbi12i 733	. . . . . . 7 |- ( ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) <-> ( ( h |` ( dom h i^i dom g ) ) Fn ( dom h i^i dom g ) /\ A. a e. ( dom h i^i dom g ) ( ( h |` ( dom h i^i dom g ) ) ` a ) = ( F ` ( ( h |` ( dom h i^i dom g ) ) |` Pred ( R , ( dom h i^i dom g ) , a ) ) ) ) )
43	29, 42	sylibr 224	. . . . . 6 |- ( ( g e. B /\ h e. B ) -> ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) )
44		wfr3g 7413	. . . . . 6 |- ( ( ( R We ( dom g i^i dom h ) /\ R Se ( dom g i^i dom h ) ) /\ ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( g |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) /\ ( ( h |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ A. a e. ( dom g i^i dom h ) ( ( h |` ( dom g i^i dom h ) ) ` a ) = ( F ` ( ( h |` ( dom g i^i dom h ) ) |` Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
45	26, 27, 43, 44	syl3anc 1326	. . . . 5 |- ( ( g e. B /\ h e. B ) -> ( g |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
46	45	breqd 4664	. . . 4 |- ( ( g e. B /\ h e. B ) -> ( x ( g |` ( dom g i^i dom h ) ) v <-> x ( h |` ( dom g i^i dom h ) ) v ) )
47	46	biimprd 238	. . 3 |- ( ( g e. B /\ h e. B ) -> ( x ( h |` ( dom g i^i dom h ) ) v -> x ( g |` ( dom g i^i dom h ) ) v ) )
48	15	wfrlem2 7415	. . . . 5 |- ( g e. B -> Fun g )
49		funres 5929	. . . . 5 |- ( Fun g -> Fun ( g |` ( dom g i^i dom h ) ) )
50		dffun2 5898	. . . . . 6 |- ( Fun ( g |` ( dom g i^i dom h ) ) <-> ( Rel ( g |` ( dom g i^i dom h ) ) /\ A. x A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) ) )
51	50	simprbi 480	. . . . 5 |- ( Fun ( g |` ( dom g i^i dom h ) ) -> A. x A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) )
52		2sp 2056	. . . . . 6 |- ( A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) )
53	52	sps 2055	. . . . 5 |- ( A. x A. u A. v ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) )
54	48, 49, 51, 53	4syl 19	. . . 4 |- ( g e. B -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) )
55	54	adantr 481	. . 3 |- ( ( g e. B /\ h e. B ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( g |` ( dom g i^i dom h ) ) v ) -> u = v ) )
56	47, 55	sylan2d 499	. 2 |- ( ( g e. B /\ h e. B ) -> ( ( x ( g |` ( dom g i^i dom h ) ) u /\ x ( h |` ( dom g i^i dom h ) ) v ) -> u = v ) )
57	14, 56	syl5 34	1 |- ( ( g e. B /\ h e. B ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653   Se wse 5071   We wwe 5072   dom cdm 5114   |` cres 5116   Rel wrel 5119   Predcpred 5679   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

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

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-rmo 2920  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  wfrfun  7425