Theorem tfr1onlemex 6315

Description: Lemma for tfr1on 6318. (Contributed by Jim Kingdon, 16-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	|- F = recs ( G )
tfr1on.g	|- ( ph -> Fun G )
tfr1on.x	|- ( ph -> Ord X )
tfr1on.ex	|- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
tfr1onlemsucfn.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfr1onlembacc.3	|- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
tfr1onlembacc.u	|- ( ( ph /\ x e. U. X ) -> suc x e. X )
tfr1onlembacc.4	|- ( ph -> D e. X )
tfr1onlembacc.5	|- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfr1onlemex	|- ( ph -> E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) )

Distinct variable groups:   A, f, g, h, x, z   D, f, g, x   f, G, x, y   f, X, x   ph, f, g, h, x, z   y, g, z   B, f, g, h, w, z   u, B, f, w   D, h, w, z, x   u, D   h, G, z, y   u, G, w   g, X, z   ph, w   y, w

Allowed substitution hints:   ph( y, u)   A( y, w, u)   B( x, y)   D( y)   F( x, y, z, w, u, f, g, h)   G( g)   X( y, w, u, h)

Proof of Theorem tfr1onlemex

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . 4 |- F = recs ( G )
2		tfr1on.g	. . . 4 |- ( ph -> Fun G )
3		tfr1on.x	. . . 4 |- ( ph -> Ord X )
4		tfr1on.ex	. . . 4 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
5		tfr1onlemsucfn.1	. . . 4 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
6		tfr1onlembacc.3	. . . 4 |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
7		tfr1onlembacc.u	. . . 4 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
8		tfr1onlembacc.4	. . . 4 |- ( ph -> D e. X )
9		tfr1onlembacc.5	. . . 4 |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembex 6313	. . 3 |- ( ph -> B e. _V )
11		uniexg 4417	. . 3 |- ( B e. _V -> U. B e. _V )
12	10, 11	syl 14	. 2 |- ( ph -> U. B e. _V )
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembfn 6312	. . 3 |- ( ph -> U. B Fn D )
14	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlemubacc 6314	. . 3 |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )
15	13, 14	jca 304	. 2 |- ( ph -> ( U. B Fn D /\ A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) )
16		fneq1 5276	. . . 4 |- ( f = U. B -> ( f Fn D <-> U. B Fn D ) )
17		fveq1 5485	. . . . . 6 |- ( f = U. B -> ( f ` u ) = ( U. B ` u ) )
18		reseq1 4878	. . . . . . 7 |- ( f = U. B -> ( f |` u ) = ( U. B |` u ) )
19	18	fveq2d 5490	. . . . . 6 |- ( f = U. B -> ( G ` ( f |` u ) ) = ( G ` ( U. B |` u ) ) )
20	17, 19	eqeq12d 2180	. . . . 5 |- ( f = U. B -> ( ( f ` u ) = ( G ` ( f |` u ) ) <-> ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) )
21	20	ralbidv 2466	. . . 4 |- ( f = U. B -> ( A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) <-> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) )
22	16, 21	anbi12d 465	. . 3 |- ( f = U. B -> ( ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) <-> ( U. B Fn D /\ A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) ) )
23	22	spcegv 2814	. 2 |- ( U. B e. _V -> ( ( U. B Fn D /\ A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) -> E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) ) )
24	12, 15, 23	sylc 62	1 |- ( ph -> E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   u. cun 3114   {csn 3576   <.cop 3579   U.cuni 3789   Ord word 4340   suc csuc 4343   |` cres 4606   Fun wfun 5182   Fn wfn 5183   ` cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273

This theorem is referenced by:  tfr1onlemaccex  6316