Theorem tfrlem5 7476

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 3203	. . 3 |- g e. _V
3	1, 2	tfrlem3a 7473	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
4		vex 3203	. . 3 |- h e. _V
5	1, 4	tfrlem3a 7473	. 2 |- ( h e. A <-> E. w e. On ( h Fn w /\ A. a e. w ( h ` a ) = ( F ` ( h |` a ) ) ) )
6		reeanv 3107	. . 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 1128	. . . . . . . . 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 ) ) -> g Fn z )
8		simp3l 1089	. . . . . . . . 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 g u )
9		fnbr 5993	. . . . . . . . 9 |- ( ( g Fn z /\ x g u ) -> x e. z )
10	7, 8, 9	syl2anc 693	. . . . . . . 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 )
11		simp2rl 1130	. . . . . . . . 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 ) ) -> h Fn w )
12		simp3r 1090	. . . . . . . . 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 h v )
13		fnbr 5993	. . . . . . . . 9 |- ( ( h Fn w /\ x h v ) -> x e. w )
14	11, 12, 13	syl2anc 693	. . . . . . . 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. w )
15	10, 14	elind 3798	. . . . . . 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 ) ) -> x e. ( z i^i w ) )
16		onin 5754	. . . . . . . . 9 |- ( ( z e. On /\ w e. On ) -> ( z i^i w ) e. On )
17	16	3ad2ant1 1082	. . . . . . . 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 ) ) -> ( z i^i w ) e. On )
18		fnfun 5988	. . . . . . . . . 10 |- ( g Fn z -> Fun g )
19	7, 18	syl 17	. . . . . . . . 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 )
20		inss1 3833	. . . . . . . . . 10 |- ( z i^i w ) C_ z
21		fndm 5990	. . . . . . . . . . 11 |- ( g Fn z -> dom g = z )
22	7, 21	syl 17	. . . . . . . . . 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 ) ) -> dom g = z )
23	20, 22	syl5sseqr 3654	. . . . . . . . 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 ) C_ dom g )
24	19, 23	jca 554	. . . . . . . 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 ) ) -> ( Fun g /\ ( z i^i w ) C_ dom g ) )
25		fnfun 5988	. . . . . . . . . 10 |- ( h Fn w -> Fun h )
26	11, 25	syl 17	. . . . . . . . 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 )
27		inss2 3834	. . . . . . . . . 10 |- ( z i^i w ) C_ w
28		fndm 5990	. . . . . . . . . . 11 |- ( h Fn w -> dom h = w )
29	11, 28	syl 17	. . . . . . . . . 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 ) ) -> dom h = w )
30	27, 29	syl5sseqr 3654	. . . . . . . . 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 ) C_ dom h )
31	26, 30	jca 554	. . . . . . . 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 ) ) -> ( Fun h /\ ( z i^i w ) C_ dom h ) )
32		simp2lr 1129	. . . . . . . . 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 ( g ` a ) = ( F ` ( g |` a ) ) )
33		ssralv 3666	. . . . . . . . 9 |- ( ( 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 ) ) ) )
34	20, 32, 33	mpsyl 68	. . . . . . . 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 ) = ( F ` ( g |` a ) ) )
35		simp2rr 1131	. . . . . . . . 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. w ( h ` a ) = ( F ` ( h |` a ) ) )
36		ssralv 3666	. . . . . . . . 9 |- ( ( 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 ) ) ) )
37	27, 35, 36	mpsyl 68	. . . . . . . 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 ) ( h ` a ) = ( F ` ( h |` a ) ) )
38	17, 24, 31, 34, 37	tfrlem1 7472	. . . . . . 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 ) ) -> A. a e. ( z i^i w ) ( g ` a ) = ( h ` a ) )
39		fveq2 6191	. . . . . . . . 9 |- ( a = x -> ( g ` a ) = ( g ` x ) )
40		fveq2 6191	. . . . . . . . 9 |- ( a = x -> ( h ` a ) = ( h ` x ) )
41	39, 40	eqeq12d 2637	. . . . . . . 8 |- ( a = x -> ( ( g ` a ) = ( h ` a ) <-> ( g ` x ) = ( h ` x ) ) )
42	41	rspcv 3305	. . . . . . 7 |- ( x e. ( z i^i w ) -> ( A. a e. ( z i^i w ) ( g ` a ) = ( h ` a ) -> ( g ` x ) = ( h ` x ) ) )
43	15, 38, 42	sylc 65	. . . . . 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 ) ) -> ( g ` x ) = ( h ` x ) )
44		funbrfv 6234	. . . . . . 7 |- ( Fun g -> ( x g u -> ( g ` x ) = u ) )
45	19, 8, 44	sylc 65	. . . . . 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 ) ) -> ( g ` x ) = u )
46		funbrfv 6234	. . . . . . 7 |- ( Fun h -> ( x h v -> ( h ` x ) = v ) )
47	26, 12, 46	sylc 65	. . . . . 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 ) ) -> ( h ` x ) = v )
48	43, 45, 47	3eqtr3d 2664	. . . . 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 )
49	48	3exp 1264	. . . 4 |- ( ( 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	rexlimivv 3036	. . 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 ) )
51	6, 50	sylbir 225	. 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 ) )
52	3, 5, 51	syl2anb 496	1 |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   |` cres 5116   Oncon0 5723   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  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  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-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  tfrlem7  7479