Theorem tfrlem5 5930

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 2560	. . 3 |- g e. _V
3	1, 2	tfrlem3a 5925	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
4		vex 2560	. . 3 |- h e. _V
5	1, 4	tfrlem3a 5925	. 2 |- ( h e. A <-> E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) )
6		reeanv 2479	. . 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 971	. . . . . . . . . 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 932	. . . . . . . . . 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 5001	. . . . . . . . . 10 |- ( ( g Fn z /\ x g u ) -> x e. z )
10	7, 8, 9	syl2anc 391	. . . . . . . . 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 973	. . . . . . . . . 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 933	. . . . . . . . . 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 5001	. . . . . . . . . 10 |- ( ( h Fn w /\ x h v ) -> x e. w )
14	11, 12, 13	syl2anc 391	. . . . . . . . 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 3126	. . . . . . . . 9 |- ( x e. ( z i^i w ) <-> ( x e. z /\ x e. w ) )
16	10, 14, 15	sylanbrc 394	. . . . . . . 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 4123	. . . . . . . . . 10 |- ( ( z e. On /\ w e. On ) -> ( z i^i w ) e. On )
18	17	3ad2ant1 925	. . . . . . . . 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 4996	. . . . . . . . . . 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 3157	. . . . . . . . . . 11 |- ( z i^i w ) C_ z
22		fndm 4998	. . . . . . . . . . . 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 2994	. . . . . . . . . 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 290	. . . . . . . . 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 4996	. . . . . . . . . . 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 3158	. . . . . . . . . . 11 |- ( z i^i w ) C_ w
29		fndm 4998	. . . . . . . . . . . 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 2994	. . . . . . . . . 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 290	. . . . . . . . 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 972	. . . . . . . . . 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 3004	. . . . . . . . . 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 59	. . . . . . . . 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 974	. . . . . . . . . 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 3004	. . . . . . . . . 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 59	. . . . . . . . 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 5923	. . . . . . . 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 5178	. . . . . . . . . 10 |- ( a = x -> ( g ` a ) = ( g ` x ) )
41		fveq2 5178	. . . . . . . . . 10 |- ( a = x -> ( h ` a ) = ( h ` x ) )
42	40, 41	eqeq12d 2054	. . . . . . . . 9 |- ( a = x -> ( ( g ` a ) = ( h ` a ) <-> ( g ` x ) = ( h ` x ) ) )
43	42	rspcv 2652	. . . . . . . 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 56	. . . . . . 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 5212	. . . . . . . 8 |- ( Fun g -> ( x g u -> ( g ` x ) = u ) )
46	20, 8, 45	sylc 56	. . . . . . 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 5212	. . . . . . . 8 |- ( Fun h -> ( x h v -> ( h ` x ) = v ) )
48	27, 12, 47	sylc 56	. . . . . . 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 2080	. . . . . 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 1103	. . . . 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 2433	. . . 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 2427	. . 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 125	. 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 275	1 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   i^i cin 2916   C_ wss 2917   class class class wbr 3764   Oncon0 4100   dom cdm 4345   |` cres 4347   Fun wfun 4896   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by:  tfrlem7  5933  tfrexlem  5948