Theorem tfrlemi1 5946

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 103	. . . . . . 7 |- ( ( z = w /\ g = k ) -> g = k )
2		simpl 102	. . . . . . 7 |- ( ( z = w /\ g = k ) -> z = w )
3	1, 2	fneq12d 4991	. . . . . 6 |- ( ( z = w /\ g = k ) -> ( g Fn z <-> k Fn w ) )
4	1	fveq1d 5180	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( g ` u ) = ( k ` u ) )
5	1	reseq1d 4611	. . . . . . . . 9 |- ( ( z = w /\ g = k ) -> ( g |` u ) = ( k |` u ) )
6	5	fveq2d 5182	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( F ` ( g |` u ) ) = ( F ` ( k |` u ) ) )
7	4, 6	eqeq12d 2054	. . . . . . 7 |- ( ( z = w /\ g = k ) -> ( ( g ` u ) = ( F ` ( g |` u ) ) <-> ( k ` u ) = ( F ` ( k |` u ) ) ) )
8	2, 7	raleqbidv 2517	. . . . . 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 442	. . . . 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 1804	. . . 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 219	. . 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 4988	. . . . . 6 |- ( z = C -> ( g Fn z <-> g Fn C ) )
13		raleq 2505	. . . . . 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 442	. . . . 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 1706	. . . 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 219	. . 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 2396	. . . 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 5926	. . . . . . . 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 5178	. . . . . . . . . . . . 13 |- ( x = z -> ( F ` x ) = ( F ` z ) )
22	21	eleq1d 2106	. . . . . . . . . . . 12 |- ( x = z -> ( ( F ` x ) e. _V <-> ( F ` z ) e. _V ) )
23	22	anbi2d 437	. . . . . . . . . . 11 |- ( x = z -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` z ) e. _V ) ) )
24	23	cbvalv 1794	. . . . . . . . . 10 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. z ( Fun F /\ ( F ` z ) e. _V ) )
25	20, 24	sylib 127	. . . . . . . . 9 |- ( ph -> A. z ( Fun F /\ ( F ` z ) e. _V ) )
26	25	adantr 261	. . . . . . . 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 103	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> k = f )
28		simplr 482	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> w = v )
29	27, 28	fneq12d 4991	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k Fn w <-> f Fn v ) )
30	27	eleq1d 2106	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k e. A <-> f e. A ) )
31		simpll 481	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> t = h )
32	27	fveq2d 5182	. . . . . . . . . . . . . . . 16 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( F ` k ) = ( F ` f ) )
33	28, 32	opeq12d 3557	. . . . . . . . . . . . . . 15 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> <. w , ( F ` k ) >. = <. v , ( F ` f ) >. )
34	33	sneqd 3388	. . . . . . . . . . . . . 14 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> { <. w , ( F ` k ) >. } = { <. v , ( F ` f ) >. } )
35	27, 34	uneq12d 3098	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k u. { <. w , ( F ` k ) >. } ) = ( f u. { <. v , ( F ` f ) >. } ) )
36	31, 35	eqeq12d 2054	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( t = ( k u. { <. w , ( F ` k ) >. } ) <-> h = ( f u. { <. v , ( F ` f ) >. } ) ) )
37	29, 30, 36	3anbi123d 1207	. . . . . . . . . . 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 1804	. . . . . . . . . 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 2540	. . . . . . . . 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 2161	. . . . . . . 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 102	. . . . . . . . 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 262	. . . . . . . 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 103	. . . . . . . . . 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 103	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> k = f )
45		simpl 102	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> w = v )
46	44, 45	fneq12d 4991	. . . . . . . . . . . . 13 |- ( ( w = v /\ k = f ) -> ( k Fn w <-> f Fn v ) )
47		simplr 482	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> k = f )
48		simpr 103	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> u = y )
49	47, 48	fveq12d 5184	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k ` u ) = ( f ` y ) )
50	47, 48	reseq12d 4613	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k |` u ) = ( f |` y ) )
51	50	fveq2d 5182	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( F ` ( k |` u ) ) = ( F ` ( f |` y ) ) )
52	49, 51	eqeq12d 2054	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( ( k ` u ) = ( F ` ( k |` u ) ) <-> ( f ` y ) = ( F ` ( f |` y ) ) ) )
53		simpll 481	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> w = v )
54	52, 53	cbvraldva2 2537	. . . . . . . . . . . . 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 442	. . . . . . . . . . . 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 1804	. . . . . . . . . . 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 2533	. . . . . . . . . 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 127	. . . . . . . . 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 262	. . . . . . . 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 5945	. . . . . . 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 357	. . . . . 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 109	. . . . 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 23	. . . 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 141	. . 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 4309	. 2 |- ( C e. On -> ( ph -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
66	65	impcom 116	1 |- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557   u. cun 2915   {csn 3375   <.cop 3378   Oncon0 4100   |` 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-in1 544  ax-in2 545  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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  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-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920

This theorem is referenced by:  tfrlemi14d  5947  tfrexlem  5948