Theorem tfrlemi1 5969

Description: We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that F is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlemisucfn.2	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )

Assertion

Ref	Expression
tfrlemi1	|- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )

Distinct variable groups:   f, g, u, x, y, A   f, F, g, u, x, y   ph, y   C, g, u   ph, f

Allowed substitution hints:   ph( x, u, g)   C( x, y, f)

Proof of Theorem tfrlemi1

Dummy variables e h k t v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 108	. . . . . . 7 |- ( ( z = w /\ g = k ) -> g = k )
2		simpl 107	. . . . . . 7 |- ( ( z = w /\ g = k ) -> z = w )
3	1, 2	fneq12d 5011	. . . . . 6 |- ( ( z = w /\ g = k ) -> ( g Fn z <-> k Fn w ) )
4	1	fveq1d 5200	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( g ` u ) = ( k ` u ) )
5	1	reseq1d 4629	. . . . . . . . 9 |- ( ( z = w /\ g = k ) -> ( g |` u ) = ( k |` u ) )
6	5	fveq2d 5202	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( F ` ( g |` u ) ) = ( F ` ( k |` u ) ) )
7	4, 6	eqeq12d 2095	. . . . . . 7 |- ( ( z = w /\ g = k ) -> ( ( g ` u ) = ( F ` ( g |` u ) ) <-> ( k ` u ) = ( F ` ( k |` u ) ) ) )
8	2, 7	raleqbidv 2561	. . . . . 6 |- ( ( z = w /\ g = k ) -> ( A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) <-> A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) )
9	3, 8	anbi12d 456	. . . . 5 |- ( ( z = w /\ g = k ) -> ( ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) <-> ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) )
10	9	cbvexdva 1845	. . . 4 |- ( z = w -> ( E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) <-> E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) )
11	10	imbi2d 228	. . 3 |- ( z = w -> ( ( ph -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) <-> ( ph -> E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) ) )
12		fneq2 5008	. . . . . 6 |- ( z = C -> ( g Fn z <-> g Fn C ) )
13		raleq 2549	. . . . . 6 |- ( z = C -> ( A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) <-> A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )
14	12, 13	anbi12d 456	. . . . 5 |- ( z = C -> ( ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) <-> ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
15	14	exbidv 1746	. . . 4 |- ( z = C -> ( E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) <-> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
16	15	imbi2d 228	. . 3 |- ( z = C -> ( ( ph -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) <-> ( ph -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) ) )
17		r19.21v 2438	. . . 4 |- ( A. w e. z ( ph -> E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) <-> ( ph -> A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) )
18		tfrlemisucfn.1	. . . . . . . . 9 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	tfrlem3 5949	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. e e. z ( g ` e ) = ( F ` ( g |` e ) ) ) }
20		tfrlemisucfn.2	. . . . . . . . . 10 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
21		fveq2 5198	. . . . . . . . . . . . 13 |- ( x = z -> ( F ` x ) = ( F ` z ) )
22	21	eleq1d 2147	. . . . . . . . . . . 12 |- ( x = z -> ( ( F ` x ) e. _V <-> ( F ` z ) e. _V ) )
23	22	anbi2d 451	. . . . . . . . . . 11 |- ( x = z -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` z ) e. _V ) ) )
24	23	cbvalv 1835	. . . . . . . . . 10 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. z ( Fun F /\ ( F ` z ) e. _V ) )
25	20, 24	sylib 120	. . . . . . . . 9 |- ( ph -> A. z ( Fun F /\ ( F ` z ) e. _V ) )
26	25	adantr 270	. . . . . . . 8 |- ( ( ph /\ ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) ) -> A. z ( Fun F /\ ( F ` z ) e. _V ) )
27		simpr 108	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> k = f )
28		simplr 496	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> w = v )
29	27, 28	fneq12d 5011	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k Fn w <-> f Fn v ) )
30	27	eleq1d 2147	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k e. A <-> f e. A ) )
31		simpll 495	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> t = h )
32	27	fveq2d 5202	. . . . . . . . . . . . . . . 16 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( F ` k ) = ( F ` f ) )
33	28, 32	opeq12d 3578	. . . . . . . . . . . . . . 15 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> <. w , ( F ` k ) >. = <. v , ( F ` f ) >. )
34	33	sneqd 3411	. . . . . . . . . . . . . 14 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> { <. w , ( F ` k ) >. } = { <. v , ( F ` f ) >. } )
35	27, 34	uneq12d 3127	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k u. { <. w , ( F ` k ) >. } ) = ( f u. { <. v , ( F ` f ) >. } ) )
36	31, 35	eqeq12d 2095	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( t = ( k u. { <. w , ( F ` k ) >. } ) <-> h = ( f u. { <. v , ( F ` f ) >. } ) ) )
37	29, 30, 36	3anbi123d 1243	. . . . . . . . . . 11 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( ( k Fn w /\ k e. A /\ t = ( k u. { <. w , ( F ` k ) >. } ) ) <-> ( f Fn v /\ f e. A /\ h = ( f u. { <. v , ( F ` f ) >. } ) ) ) )
38	37	cbvexdva 1845	. . . . . . . . . 10 |- ( ( t = h /\ w = v ) -> ( E. k ( k Fn w /\ k e. A /\ t = ( k u. { <. w , ( F ` k ) >. } ) ) <-> E. f ( f Fn v /\ f e. A /\ h = ( f u. { <. v , ( F ` f ) >. } ) ) ) )
39	38	cbvrexdva 2584	. . . . . . . . 9 |- ( t = h -> ( E. w e. z E. k ( k Fn w /\ k e. A /\ t = ( k u. { <. w , ( F ` k ) >. } ) ) <-> E. v e. z E. f ( f Fn v /\ f e. A /\ h = ( f u. { <. v , ( F ` f ) >. } ) ) ) )
40	39	cbvabv 2202	. . . . . . . 8 |- { t | E. w e. z E. k ( k Fn w /\ k e. A /\ t = ( k u. { <. w , ( F ` k ) >. } ) ) } = { h | E. v e. z E. f ( f Fn v /\ f e. A /\ h = ( f u. { <. v , ( F ` f ) >. } ) ) }
41		simpl 107	. . . . . . . . 9 |- ( ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) -> z e. On )
42	41	adantl 271	. . . . . . . 8 |- ( ( ph /\ ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) ) -> z e. On )
43		simpr 108	. . . . . . . . . 10 |- ( ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) -> A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) )
44		simpr 108	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> k = f )
45		simpl 107	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> w = v )
46	44, 45	fneq12d 5011	. . . . . . . . . . . . 13 |- ( ( w = v /\ k = f ) -> ( k Fn w <-> f Fn v ) )
47		simplr 496	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> k = f )
48		simpr 108	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> u = y )
49	47, 48	fveq12d 5204	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k ` u ) = ( f ` y ) )
50	47, 48	reseq12d 4631	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k |` u ) = ( f |` y ) )
51	50	fveq2d 5202	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( F ` ( k |` u ) ) = ( F ` ( f |` y ) ) )
52	49, 51	eqeq12d 2095	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( ( k ` u ) = ( F ` ( k |` u ) ) <-> ( f ` y ) = ( F ` ( f |` y ) ) ) )
53		simpll 495	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> w = v )
54	52, 53	cbvraldva2 2581	. . . . . . . . . . . . 13 |- ( ( w = v /\ k = f ) -> ( A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) <-> A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) )
55	46, 54	anbi12d 456	. . . . . . . . . . . 12 |- ( ( w = v /\ k = f ) -> ( ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) <-> ( f Fn v /\ A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) ) )
56	55	cbvexdva 1845	. . . . . . . . . . 11 |- ( w = v -> ( E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) <-> E. f ( f Fn v /\ A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) ) )
57	56	cbvralv 2577	. . . . . . . . . 10 |- ( A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) <-> A. v e. z E. f ( f Fn v /\ A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) )
58	43, 57	sylib 120	. . . . . . . . 9 |- ( ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) -> A. v e. z E. f ( f Fn v /\ A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) )
59	58	adantl 271	. . . . . . . 8 |- ( ( ph /\ ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) ) -> A. v e. z E. f ( f Fn v /\ A. y e. v ( f ` y ) = ( F ` ( f |` y ) ) ) )
60	19, 26, 40, 42, 59	tfrlemiex 5968	. . . . . . 7 |- ( ( ph /\ ( z e. On /\ A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) )
61	60	expr 367	. . . . . 6 |- ( ( ph /\ z e. On ) -> ( A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
62	61	expcom 114	. . . . 5 |- ( z e. On -> ( ph -> ( A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) ) )
63	62	a2d 26	. . . 4 |- ( z e. On -> ( ( ph -> A. w e. z E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) -> ( ph -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) ) )
64	17, 63	syl5bi 150	. . 3 |- ( z e. On -> ( A. w e. z ( ph -> E. k ( k Fn w /\ A. u e. w ( k ` u ) = ( F ` ( k |` u ) ) ) ) -> ( ph -> E. g ( g Fn z /\ A. u e. z ( g ` u ) = ( F ` ( g |` u ) ) ) ) ) )
65	11, 16, 64	tfis3 4327	. 2 |- ( C e. On -> ( ph -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
66	65	impcom 123	1 |- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   _Vcvv 2601   u. cun 2971   {csn 3398   <.cop 3401   Oncon0 4118   |` 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-in1 576  ax-in2 577  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-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  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-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943

This theorem is referenced by:  tfrlemi14d  5970  tfrexlem  5971