Theorem tfr1onlemex 6493

Description: Lemma for tfr1on 6496. (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 6491	. . 3 |- ( ph -> B e. _V )
11		uniexg 4530	. . 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 6490	. . 3 |- ( ph -> U. B Fn D )
14	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlemubacc 6492	. . 3 |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )
15	13, 14	jca 306	. 2 |- ( ph -> ( U. B Fn D /\ A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) )
16		fneq1 5409	. . . 4 |- ( f = U. B -> ( f Fn D <-> U. B Fn D ) )
17		fveq1 5626	. . . . . 6 |- ( f = U. B -> ( f ` u ) = ( U. B ` u ) )
18		reseq1 4999	. . . . . . 7 |- ( f = U. B -> ( f |` u ) = ( U. B |` u ) )
19	18	fveq2d 5631	. . . . . 6 |- ( f = U. B -> ( G ` ( f |` u ) ) = ( G ` ( U. B |` u ) ) )
20	17, 19	eqeq12d 2244	. . . . 5 |- ( f = U. B -> ( ( f ` u ) = ( G ` ( f |` u ) ) <-> ( U. B ` u ) = ( G ` ( U. B |` u ) ) ) )
21	20	ralbidv 2530	. . . 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 473	. . 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 2891	. 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 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2799   u. cun 3195   {csn 3666   <.cop 3669   U.cuni 3888   Ord word 4453   suc csuc 4456   |` cres 4721   Fun wfun 5312   Fn wfn 5313   ` cfv 5318  recscrecs 6450

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-recs 6451

This theorem is referenced by:  tfr1onlemaccex  6494