Theorem tfrlemibacc 6414

Description: Each element of B is an acceptable function. Lemma for tfrlemi1 6420. (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 1008	. . . . . . 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 4432	. . . . . . . . 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 1006	. . . . . . . 8 |- ( ( ( ph /\ z e. x ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) ) -> g Fn z )
12		simpr2 1007	. . . . . . . 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 6413	. . . . . . 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 2282	. . . . . 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 1842	. . . 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 2623	. . 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 3267	. 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 3239	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   _Vcvv 2772   u. cun 3164   C_ wss 3166   {csn 3633   <.cop 3636   Oncon0 4411   |` cres 4678   Fun wfun 5266   Fn wfn 5267   ` cfv 5272

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  tfrlemibfn  6416  tfrlemiubacc  6418