Theorem tfrlemi1 6420

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 5367	. . . . . 6 |- ( ( z = w /\ g = k ) -> ( g Fn z <-> k Fn w ) )
4	1	fveq1d 5580	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( g ` u ) = ( k ` u ) )
5	1	reseq1d 4959	. . . . . . . . 9 |- ( ( z = w /\ g = k ) -> ( g |` u ) = ( k |` u ) )
6	5	fveq2d 5582	. . . . . . . 8 |- ( ( z = w /\ g = k ) -> ( F ` ( g |` u ) ) = ( F ` ( k |` u ) ) )
7	4, 6	eqeq12d 2220	. . . . . . 7 |- ( ( z = w /\ g = k ) -> ( ( g ` u ) = ( F ` ( g |` u ) ) <-> ( k ` u ) = ( F ` ( k |` u ) ) ) )
8	2, 7	raleqbidv 2718	. . . . . 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 1953	. . . 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 5364	. . . . . 6 |- ( z = C -> ( g Fn z <-> g Fn C ) )
13		raleq 2702	. . . . . 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 1848	. . . 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 2583	. . . 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 6399	. . . . . . . 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 5578	. . . . . . . . . . . . 13 |- ( x = z -> ( F ` x ) = ( F ` z ) )
22	21	eleq1d 2274	. . . . . . . . . . . 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 1941	. . . . . . . . . 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 5367	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k Fn w <-> f Fn v ) )
30	27	eleq1d 2274	. . . . . . . . . . . 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 5582	. . . . . . . . . . . . . . . 16 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( F ` k ) = ( F ` f ) )
33	28, 32	opeq12d 3827	. . . . . . . . . . . . . . 15 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> <. w , ( F ` k ) >. = <. v , ( F ` f ) >. )
34	33	sneqd 3646	. . . . . . . . . . . . . 14 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> { <. w , ( F ` k ) >. } = { <. v , ( F ` f ) >. } )
35	27, 34	uneq12d 3328	. . . . . . . . . . . . 13 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( k u. { <. w , ( F ` k ) >. } ) = ( f u. { <. v , ( F ` f ) >. } ) )
36	31, 35	eqeq12d 2220	. . . . . . . . . . . 12 |- ( ( ( t = h /\ w = v ) /\ k = f ) -> ( t = ( k u. { <. w , ( F ` k ) >. } ) <-> h = ( f u. { <. v , ( F ` f ) >. } ) ) )
37	29, 30, 36	3anbi123d 1325	. . . . . . . . . . 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 1953	. . . . . . . . . 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 2748	. . . . . . . . 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 2330	. . . . . . . 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 5367	. . . . . . . . . . . . 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 5585	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k ` u ) = ( f ` y ) )
50	47, 48	reseq12d 4961	. . . . . . . . . . . . . . . 16 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( k |` u ) = ( f |` y ) )
51	50	fveq2d 5582	. . . . . . . . . . . . . . 15 |- ( ( ( w = v /\ k = f ) /\ u = y ) -> ( F ` ( k |` u ) ) = ( F ` ( f |` y ) ) )
52	49, 51	eqeq12d 2220	. . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . 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 1953	. . . . . . . . . . 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 2738	. . . . . . . . . 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 6419	. . . . . . 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 4635	. 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 981   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   _Vcvv 2772   u. cun 3164   {csn 3633   <.cop 3636   Oncon0 4411   |` cres 4678   Fun wfun 5266   Fn wfn 5267   ` cfv 5272

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-recs 6393

This theorem is referenced by:  tfrlemi14d  6421  tfrexlem  6422