Theorem tfrlem5 6372

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 2766	. . 3 |- g e. _V
3	1, 2	tfrlem3a 6368	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
4		vex 2766	. . 3 |- h e. _V
5	1, 4	tfrlem3a 6368	. 2 |- ( h e. A <-> E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) )
6		reeanv 2667	. . 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 5558	. . . . . . . . 9 |- ( a = x -> ( g ` a ) = ( g ` x ) )
8		fveq2 5558	. . . . . . . . 9 |- ( a = x -> ( h ` a ) = ( h ` x ) )
9	7, 8	eqeq12d 2211	. . . . . . . 8 |- ( a = x -> ( ( g ` a ) = ( h ` a ) <-> ( g ` x ) = ( h ` x ) ) )
10		onin 4421	. . . . . . . . . 10 |- ( ( z e. On /\ w e. On ) -> ( z i^i w ) e. On )
11	10	3ad2ant1 1020	. . . . . . . . 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 1066	. . . . . . . . . . 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 5355	. . . . . . . . . . 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 3383	. . . . . . . . . . 11 |- ( z i^i w ) C_ z
16		fndm 5357	. . . . . . . . . . . 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 3234	. . . . . . . . . 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 306	. . . . . . . . 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 1068	. . . . . . . . . . 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 5355	. . . . . . . . . . 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 3384	. . . . . . . . . . 11 |- ( z i^i w ) C_ w
24		fndm 5357	. . . . . . . . . . . 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 3234	. . . . . . . . . 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 306	. . . . . . . . 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 1067	. . . . . . . . . 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 3247	. . . . . . . . . 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 1069	. . . . . . . . . 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 3247	. . . . . . . . . 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 6366	. . . . . . . 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 1027	. . . . . . . . . 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 5360	. . . . . . . . . 10 |- ( ( g Fn z /\ x g u ) -> x e. z )
37	12, 35, 36	syl2anc 411	. . . . . . . . 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 1028	. . . . . . . . . 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 5360	. . . . . . . . . 10 |- ( ( h Fn w /\ x h v ) -> x e. w )
40	20, 38, 39	syl2anc 411	. . . . . . . . 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 3346	. . . . . . . . 9 |- ( x e. ( z i^i w ) <-> ( x e. z /\ x e. w ) )
42	37, 40, 41	sylanbrc 417	. . . . . . . 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 2873	. . . . . . 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 5599	. . . . . . . 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 5599	. . . . . . . 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 2237	. . . . . 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 1204	. . . . 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 2614	. . . 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 2608	. . 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 135	. 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 291	1 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   i^i cin 3156   C_ wss 3157   class class class wbr 4033   Oncon0 4398   dom cdm 4663   |` cres 4665   Fun wfun 5252   Fn wfn 5253   ` cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  tfrlem7  6375  tfrexlem  6392