Theorem tfrlem5 5953

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 2604	. . 3 |- g e. _V
3	1, 2	tfrlem3a 5948	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
4		vex 2604	. . 3 |- h e. _V
5	1, 4	tfrlem3a 5948	. 2 |- ( h e. A <-> E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) )
6		reeanv 2523	. . 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		simp2ll 1005	. . . . . . . . . 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 ) ) -> g Fn z )
8		simp3l 966	. . . . . . . . . 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 )
9		fnbr 5021	. . . . . . . . . 10 |- ( ( g Fn z /\ x g u ) -> x e. z )
10	7, 8, 9	syl2anc 403	. . . . . . . . 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 )
11		simp2rl 1007	. . . . . . . . . 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 ) ) -> h Fn w )
12		simp3r 967	. . . . . . . . . 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 )
13		fnbr 5021	. . . . . . . . . 10 |- ( ( h Fn w /\ x h v ) -> x e. w )
14	11, 12, 13	syl2anc 403	. . . . . . . . 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 )
15		elin 3155	. . . . . . . . 9 |- ( x e. ( z i^i w ) <-> ( x e. z /\ x e. w ) )
16	10, 14, 15	sylanbrc 408	. . . . . . . 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 ) )
17		onin 4141	. . . . . . . . . 10 |- ( ( z e. On /\ w e. On ) -> ( z i^i w ) e. On )
18	17	3ad2ant1 959	. . . . . . . . 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 )
19		fnfun 5016	. . . . . . . . . . 11 |- ( g Fn z -> Fun g )
20	7, 19	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 )
21		inss1 3186	. . . . . . . . . . 11 |- ( z i^i w ) C_ z
22		fndm 5018	. . . . . . . . . . . 12 |- ( g Fn z -> dom g = z )
23	7, 22	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 )
24	21, 23	syl5sseqr 3048	. . . . . . . . . 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 )
25	20, 24	jca 300	. . . . . . . . 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 ) )
26		fnfun 5016	. . . . . . . . . . 11 |- ( h Fn w -> Fun h )
27	11, 26	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 )
28		inss2 3187	. . . . . . . . . . 11 |- ( z i^i w ) C_ w
29		fndm 5018	. . . . . . . . . . . 12 |- ( h Fn w -> dom h = w )
30	11, 29	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 )
31	28, 30	syl5sseqr 3048	. . . . . . . . . 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 )
32	27, 31	jca 300	. . . . . . . . 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 ) )
33		simp2lr 1006	. . . . . . . . . 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 ) ) )
34		ssralv 3058	. . . . . . . . . 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 ) ) ) )
35	21, 33, 34	mpsyl 64	. . . . . . . . 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 ) ) )
36		simp2rr 1008	. . . . . . . . . 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 ) ) )
37		ssralv 3058	. . . . . . . . . 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 ) ) ) )
38	28, 36, 37	mpsyl 64	. . . . . . . . 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 ) ) )
39	18, 25, 32, 35, 38	tfrlem1 5946	. . . . . . . 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 ) )
40		fveq2 5198	. . . . . . . . . 10 |- ( a = x -> ( g ` a ) = ( g ` x ) )
41		fveq2 5198	. . . . . . . . . 10 |- ( a = x -> ( h ` a ) = ( h ` x ) )
42	40, 41	eqeq12d 2095	. . . . . . . . 9 |- ( a = x -> ( ( g ` a ) = ( h ` a ) <-> ( g ` x ) = ( h ` x ) ) )
43	42	rspcv 2697	. . . . . . . 8 |- ( x e. ( z i^i w ) -> ( A. a e. ( z i^i w ) ( g ` a ) = ( h ` a ) -> ( g ` x ) = ( h ` x ) ) )
44	16, 39, 43	sylc 61	. . . . . . 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 ) )
45		funbrfv 5233	. . . . . . . 8 |- ( Fun g -> ( x g u -> ( g ` x ) = u ) )
46	20, 8, 45	sylc 61	. . . . . . 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 )
47		funbrfv 5233	. . . . . . . 8 |- ( Fun h -> ( x h v -> ( h ` x ) = v ) )
48	27, 12, 47	sylc 61	. . . . . . 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 )
49	44, 46, 48	3eqtr3d 2121	. . . . . 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 )
50	49	3exp 1137	. . . . 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 ) ) )
51	50	rexlimdva 2477	. . . 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 ) ) )
52	51	rexlimiv 2471	. . 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 ) )
53	6, 52	sylbir 133	. 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 ) )
54	3, 5, 53	syl2anb 285	1 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   i^i cin 2972   C_ wss 2973   class class class wbr 3785   Oncon0 4118   dom cdm 4363   |` cres 4365   Fun wfun 4916   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  tfrlem7  5956  tfrexlem  5971