Theorem tfrlemiex 6540

Description: Lemma for tfrlemi1 6541. (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 6538	. . 3 |- ( ph -> B e. _V )
7		uniexg 4542	. . 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 6537	. . 3 |- ( ph -> U. B Fn x )
10	1, 2, 3, 4, 5	tfrlemiubacc 6539	. . 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 5425	. . . 4 |- ( f = U. B -> ( f Fn x <-> U. B Fn x ) )
13		fveq1 5647	. . . . . 6 |- ( f = U. B -> ( f ` u ) = ( U. B ` u ) )
14		reseq1 5013	. . . . . . 7 |- ( f = U. B -> ( f |` u ) = ( U. B |` u ) )
15	14	fveq2d 5652	. . . . . 6 |- ( f = U. B -> ( F ` ( f |` u ) ) = ( F ` ( U. B |` u ) ) )
16	13, 15	eqeq12d 2246	. . . . 5 |- ( f = U. B -> ( ( f ` u ) = ( F ` ( f |` u ) ) <-> ( U. B ` u ) = ( F ` ( U. B |` u ) ) ) )
17	16	ralbidv 2533	. . . 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 2895	. 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 2202   {cab 2217   A.wral 2511   E.wrex 2512   _Vcvv 2803   u. cun 3199   {csn 3673   <.cop 3676   U.cuni 3898   Oncon0 4466   |` cres 4733   Fun wfun 5327   Fn wfn 5328   ` cfv 5333

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfrlemi1  6541