Theorem tfrlemiex 6575

Description: Lemma for tfrlemi1 6576. (Contributed by Jim Kingdon, 18-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 ) )
tfrlemi1.3	|- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }
tfrlemi1.4	|- ( ph -> x e. On )
tfrlemi1.5	|- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfrlemiex	|- ( ph -> E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) )

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

Allowed substitution hints:   ph( x, u, f)   B( x, y)

Proof of Theorem tfrlemiex

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . 4 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2		tfrlemisucfn.2	. . . 4 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
3		tfrlemi1.3	. . . 4 |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }
4		tfrlemi1.4	. . . 4 |- ( ph -> x e. On )
5		tfrlemi1.5	. . . 4 |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
6	1, 2, 3, 4, 5	tfrlemibex 6573	. . 3 |- ( ph -> B e. _V )
7		uniexg 4565	. . 3 |- ( B e. _V -> U. B e. _V )
8	6, 7	syl 14	. 2 |- ( ph -> U. B e. _V )
9	1, 2, 3, 4, 5	tfrlemibfn 6572	. . 3 |- ( ph -> U. B Fn x )
10	1, 2, 3, 4, 5	tfrlemiubacc 6574	. . 3 |- ( ph -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )
11	9, 10	jca 306	. 2 |- ( ph -> ( U. B Fn x /\ A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) )
12		fneq1 5449	. . . 4 |- ( f = U. B -> ( f Fn x <-> U. B Fn x ) )
13		fveq1 5674	. . . . . 6 |- ( f = U. B -> ( f ` u ) = ( U. B ` u ) )
14		reseq1 5037	. . . . . . 7 |- ( f = U. B -> ( f |` u ) = ( U. B |` u ) )
15	14	fveq2d 5679	. . . . . 6 |- ( f = U. B -> ( F ` ( f |` u ) ) = ( F ` ( U. B |` u ) ) )
16	13, 15	eqeq12d 2249	. . . . 5 |- ( f = U. B -> ( ( f ` u ) = ( F ` ( f |` u ) ) <-> ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) )
17	16	ralbidv 2544	. . . 4 |- ( f = U. B -> ( A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) <-> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) )
18	12, 17	anbi12d 473	. . 3 |- ( f = U. B -> ( ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) <-> ( U. B Fn x /\ A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) ) )
19	18	spcegv 2907	. 2 |- ( U. B e. _V -> ( ( U. B Fn x /\ A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) -> E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) ) )
20	8, 11, 19	sylc 62	1 |- ( ph -> E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   u. cun 3212   {csn 3694   <.cop 3697   U.cuni 3919   Oncon0 4489   |` cres 4756   Fun wfun 5351   Fn wfn 5352   ` cfv 5357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549

This theorem is referenced by:  tfrlemi1  6576