Theorem tfrlemibacc 6435

Description: Each element of B is an acceptable function. Lemma for tfrlemi1 6441. (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 4449	. . . . . . . . 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 6434	. . . . . . 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 2284	. . . . . 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 1843	. . . 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 2625	. . 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 3275	. 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 3247	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   u. cun 3172   C_ wss 3174   {csn 3643   <.cop 3646   Oncon0 4428   |` cres 4695   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  tfrlemibfn  6437  tfrlemiubacc  6439