Theorem tfrlem5 6204

Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem5	|- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

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

Allowed substitution hints:   A( x, y, v, u, f)

Proof of Theorem tfrlem5

Dummy variables z a w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . 3 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2		vex 2684	. . 3 |- g e. _V
3	1, 2	tfrlem3a 6200	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
4		vex 2684	. . 3 |- h e. _V
5	1, 4	tfrlem3a 6200	. 2 |- ( h e. A <-> E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) )
6		reeanv 2598	. . 3 |- ( E. z e. On E. w e. On ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) <-> ( E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) )
7		fveq2 5414	. . . . . . . . 9 |- ( a = x -> ( g ` a ) = ( g ` x ) )
8		fveq2 5414	. . . . . . . . 9 |- ( a = x -> ( h ` a ) = ( h ` x ) )
9	7, 8	eqeq12d 2152	. . . . . . . 8 |- ( a = x -> ( ( g ` a ) = ( h ` a ) <-> ( g ` x ) = ( h ` x ) ) )
10		onin 4303	. . . . . . . . . 10 |- ( ( z e. On /\ w e. On ) -> ( z i^i w ) e. On )
11	10	3ad2ant1 1002	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( z i^i w ) e. On )
12		simp2ll 1048	. . . . . . . . . . 11 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> g Fn z )
13		fnfun 5215	. . . . . . . . . . 11 |- ( g Fn z -> Fun g )
14	12, 13	syl 14	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> Fun g )
15		inss1 3291	. . . . . . . . . . 11 |- ( z i^i w ) C_ z
16		fndm 5217	. . . . . . . . . . . 12 |- ( g Fn z -> dom g = z )
17	12, 16	syl 14	. . . . . . . . . . 11 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> dom g = z )
18	15, 17	sseqtrrid 3143	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( z i^i w ) C_ dom g )
19	14, 18	jca 304	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( Fun g /\ ( z i^i w ) C_ dom g ) )
20		simp2rl 1050	. . . . . . . . . . 11 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> h Fn w )
21		fnfun 5215	. . . . . . . . . . 11 |- ( h Fn w -> Fun h )
22	20, 21	syl 14	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> Fun h )
23		inss2 3292	. . . . . . . . . . 11 |- ( z i^i w ) C_ w
24		fndm 5217	. . . . . . . . . . . 12 |- ( h Fn w -> dom h = w )
25	20, 24	syl 14	. . . . . . . . . . 11 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> dom h = w )
26	23, 25	sseqtrrid 3143	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( z i^i w ) C_ dom h )
27	22, 26	jca 304	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( Fun h /\ ( z i^i w ) C_ dom h ) )
28		simp2lr 1049	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) )
29		ssralv 3156	. . . . . . . . . 10 |- ( ( z i^i w ) C_ z -> ( A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) -> A. a e. ( z i^i w ) ( g ` a ) = ( F ` ( g |` a ) ) ) )
30	15, 28, 29	mpsyl 65	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> A. a e. ( z i^i w ) ( g ` a ) = ( F ` ( g |` a ) ) )
31		simp2rr 1051	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) )
32		ssralv 3156	. . . . . . . . . 10 |- ( ( z i^i w ) C_ w -> ( A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) -> A. a e. ( z i^i w ) ( h ` a ) = ( F ` ( h |` a ) ) ) )
33	23, 31, 32	mpsyl 65	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> A. a e. ( z i^i w ) ( h ` a ) = ( F ` ( h |` a ) ) )
34	11, 19, 27, 30, 33	tfrlem1 6198	. . . . . . . 8 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> A. a e. ( z i^i w ) ( g ` a ) = ( h ` a ) )
35		simp3l 1009	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> x g u )
36		fnbr 5220	. . . . . . . . . 10 |- ( ( g Fn z /\ x g u ) -> x e. z )
37	12, 35, 36	syl2anc 408	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> x e. z )
38		simp3r 1010	. . . . . . . . . 10 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> x h v )
39		fnbr 5220	. . . . . . . . . 10 |- ( ( h Fn w /\ x h v ) -> x e. w )
40	20, 38, 39	syl2anc 408	. . . . . . . . 9 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> x e. w )
41		elin 3254	. . . . . . . . 9 |- ( x e. ( z i^i w ) <-> ( x e. z /\ x e. w ) )
42	37, 40, 41	sylanbrc 413	. . . . . . . 8 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> x e. ( z i^i w ) )
43	9, 34, 42	rspcdva 2789	. . . . . . 7 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( g ` x ) = ( h ` x ) )
44		funbrfv 5453	. . . . . . . 8 |- ( Fun g -> ( x g u -> ( g ` x ) = u ) )
45	14, 35, 44	sylc 62	. . . . . . 7 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( g ` x ) = u )
46		funbrfv 5453	. . . . . . . 8 |- ( Fun h -> ( x h v -> ( h ` x ) = v ) )
47	22, 38, 46	sylc 62	. . . . . . 7 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> ( h ` x ) = v )
48	43, 45, 47	3eqtr3d 2178	. . . . . 6 |- ( ( ( z e. On /\ w e. On ) /\ ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) /\ ( x g u /\ x h v ) ) -> u = v )
49	48	3exp 1180	. . . . 5 |- ( ( z e. On /\ w e. On ) -> ( ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) )
50	49	rexlimdva 2547	. . . 4 |- ( z e. On -> ( E. w e. On ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) )
51	50	rexlimiv 2541	. . 3 |- ( E. z e. On E. w e. On ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) -> ( ( x g u /\ x h v ) -> u = v ) )
52	6, 51	sylbir 134	. 2 |- ( ( E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) /\ E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) ) -> ( ( x g u /\ x h v ) -> u = v ) )
53	3, 5, 52	syl2anb 289	1 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   i^i cin 3065   C_ wss 3066   class class class wbr 3924   Oncon0 4280   dom cdm 4534   |` cres 4536   Fun wfun 5112   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  tfrlem7  6207  tfrexlem  6224