Theorem tfrlemi1 6330

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 110	. . . . . . 7 |- ( ( z = w /\ g = k ) -> g = k )
2		simpl 109	. . . . . . 7 |- ( ( z = w /\ g = k ) -> z = w )
3	1, 2	fneq12d 5307	. . . . . 6 |- ( ( z = w /\ g = k ) -> ( g Fn z <-> k Fn w ) )
4	1	fveq1d 5516	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( g ` u ) = ( k ` u ) )
5	1	reseq1d 4905	. . . . . . . . 9 |- ( ( z = w /\ g = k ) -> ( g |` u ) = ( k |` u ) )
6	5	fveq2d 5518	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( F ` ( g |` u ) ) = ( F ` ( k |` u ) ) )
7	4, 6	eqeq12d 2192	. . . . . . 7 |- ( ( z = w /\ g = k ) -> ( ( g ` u ) = ( F ` ( g |` u ) ) <-> ( k ` u ) = ( F ` ( k |` u ) ) ) )
8	2, 7	raleqbidv 2684	. . . . . 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 473	. . . . 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 1929	. . . 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 230	. . 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 5304	. . . . . 6 |- ( z = C -> ( g Fn z <-> g Fn C ) )
13		raleq 2672	. . . . . 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 473	. . . . 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 1825	. . . 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 230	. . 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 2554	. . . 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 6309	. . . . . . . 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 5514	. . . . . . . . . . . . 13 |- ( x = z -> ( F ` x ) = ( F ` z ) )
22	21	eleq1d 2246	. . . . . . . . . . . 12 |- ( x = z -> ( ( F ` x ) e. _V <-> ( F ` z ) e. _V ) )
23	22	anbi2d 464	. . . . . . . . . . 11 |- ( x = z -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` z ) e. _V ) ) )
24	23	cbvalv 1917	. . . . . . . . . 10 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. z ( Fun F /\ ( F ` z ) e. _V ) )
25	20, 24	sylib 122	. . . . . . . . 9 |- ( ph -> A. z ( Fun F /\ ( F ` z ) e. _V ) )
26	25	adantr 276	. . . . . . . 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 110	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> k = f )
28		simplr 528	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> w = v )
29	27, 28	fneq12d 5307	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k Fn w <-> f Fn v ) )
30	27	eleq1d 2246	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k e. A <-> f e. A ) )
31		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> t = h )
32	27	fveq2d 5518	. . . . . . . . . . . . . . . 16 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( F ` k ) = ( F ` f ) )
33	28, 32	opeq12d 3786	. . . . . . . . . . . . . . 15 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> <. w , ( F ` k ) >. = <. v , ( F ` f ) >. )
34	33	sneqd 3605	. . . . . . . . . . . . . 14 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> { <. w , ( F ` k ) >. } = { <. v , ( F ` f ) >. } )
35	27, 34	uneq12d 3290	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k u. { <. w , ( F ` k ) >. } ) = ( f u. { <. v , ( F ` f ) >. } ) )
36	31, 35	eqeq12d 2192	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( t = ( k u. { <. w , ( F ` k ) >. } ) <-> h = ( f u. { <. v , ( F ` f ) >. } ) ) )
37	29, 30, 36	3anbi123d 1312	. . . . . . . . . . 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 1929	. . . . . . . . . 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 2713	. . . . . . . . 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 2302	. . . . . . . 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 109	. . . . . . . . 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 277	. . . . . . . 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 110	. . . . . . . . . 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 110	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> k = f )
45		simpl 109	. . . . . . . . . . . . . 14 |- ( ( w = v /\ k = f ) -> w = v )
46	44, 45	fneq12d 5307	. . . . . . . . . . . . 13 |- ( ( w = v /\ k = f ) -> ( k Fn w <-> f Fn v ) )
47		simplr 528	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> k = f )
48		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> u = y )
49	47, 48	fveq12d 5521	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k ` u ) = ( f ` y ) )
50	47, 48	reseq12d 4907	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k |` u ) = ( f |` y ) )
51	50	fveq2d 5518	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( F ` ( k |` u ) ) = ( F ` ( f |` y ) ) )
52	49, 51	eqeq12d 2192	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( ( k ` u ) = ( F ` ( k |` u ) ) <-> ( f ` y ) = ( F ` ( f |` y ) ) ) )
53		simpll 527	. . . . . . . . . . . . . 14 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> w = v )
54	52, 53	cbvraldva2 2710	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 1929	. . . . . . . . . . 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 2703	. . . . . . . . . 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 122	. . . . . . . . 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 277	. . . . . . . 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 6329	. . . . . . 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 375	. . . . . 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 116	. . . . 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	biimtrid 152	. . 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 4584	. 2 |- ( C e. On -> ( ph -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) ) )
66	65	impcom 125	1 |- ( ( ph /\ C e. On ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737   u. cun 3127   {csn 3592   <.cop 3595   Oncon0 4362   |` cres 4627   Fun wfun 5209   Fn wfn 5210   ` cfv 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-suc 4370  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-recs 6303

This theorem is referenced by:  tfrlemi14d  6331  tfrexlem  6332