Theorem tfrlemibacc 6472

Description: Each element of B is an acceptable function. Lemma for tfrlemi1 6478. (Contributed by Jim Kingdon, 14-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
tfrlemibacc	|- ( ph -> B C_ A )

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

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

Proof of Theorem tfrlemibacc

Step	Hyp	Ref	Expression
1		tfrlemi1.3	. 2 |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }
2		simpr3 1029	. . . . . . 7 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> h = ( g u. { <. z , ( F ` g ) >. } ) )
3		tfrlemisucfn.1	. . . . . . . 8 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
4		tfrlemisucfn.2	. . . . . . . . 9 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
5	4	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
6		tfrlemi1.4	. . . . . . . . . 10 |- ( ph -> x e. On )
7	6	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> x e. On )
8		simplr 528	. . . . . . . . 9 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> z e. x )
9		onelon 4475	. . . . . . . . 9 |- ( ( x e. On /\ z e. x ) -> z e. On )
10	7, 8, 9	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> z e. On )
11		simpr1 1027	. . . . . . . 8 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> g Fn z )
12		simpr2 1028	. . . . . . . 8 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> g e. A )
13	3, 5, 10, 11, 12	tfrlemisucaccv 6471	. . . . . . 7 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> ( g u. { <. z , ( F ` g ) >. } ) e. A )
14	2, 13	eqeltrd 2306	. . . . . 6 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> h e. A )
15	14	ex 115	. . . . 5 |- ( ( ph /\ z e. x ) -> ( ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) -> h e. A ) )
16	15	exlimdv 1865	. . . 4 |- ( ( ph /\ z e. x ) -> ( E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) -> h e. A ) )
17	16	rexlimdva 2648	. . 3 |- ( ph -> ( E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) -> h e. A ) )
18	17	abssdv 3298	. 2 |- ( ph -> { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) } C_ A )
19	1, 18	eqsstrid 3270	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2799   u. cun 3195   C_ wss 3197   {csn 3666   <.cop 3669   Oncon0 4454   |` cres 4721   Fun wfun 5312   Fn wfn 5313   ` cfv 5318

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-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-v 2801  df-sbc 3029  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-br 4084  df-opab 4146  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-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  tfrlemibfn  6474  tfrlemiubacc  6476