Theorem tfr1onlembacc 6573

Description: Lemma for tfr1on 6581. Each element of B is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	|- F = recs ( G )
tfr1on.g	|- ( ph -> Fun G )
tfr1on.x	|- ( ph -> Ord X )
tfr1on.ex	|- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
tfr1onlemsucfn.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfr1onlembacc.3	|- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
tfr1onlembacc.u	|- ( ( ph /\ x e. U. X ) -> suc x e. X )
tfr1onlembacc.4	|- ( ph -> D e. X )
tfr1onlembacc.5	|- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfr1onlembacc	|- ( ph -> B C_ A )

Distinct variable groups:   A, f, g, h, x, z   D, f, g, x   f, G, x, y   f, X, x   ph, f, g, h, x, z   y, g, z

Allowed substitution hints:   ph( y, w)   A( y, w)   B( x, y, z, w, f, g, h)   D( y, z, w, h)   F( x, y, z, w, f, g, h)   G( z, w, g, h)   X( y, z, w, g, h)

Proof of Theorem tfr1onlembacc

Step	Hyp	Ref	Expression
1		tfr1onlembacc.3	. 2 |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
2		simpr3 1032	. . . . . . 7 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> h = ( g u. { <. z , ( G ` g ) >. } ) )
3		tfr1on.f	. . . . . . . 8 |- F = recs ( G )
4		tfr1on.g	. . . . . . . . 9 |- ( ph -> Fun G )
5	4	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> Fun G )
6		tfr1on.x	. . . . . . . . 9 |- ( ph -> Ord X )
7	6	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> Ord X )
8		tfr1on.ex	. . . . . . . . . 10 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
9	8	3adant1r 1258	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
10	9	3adant1r 1258	. . . . . . . 8 |- ( ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
11		tfr1onlemsucfn.1	. . . . . . . 8 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
12		tfr1onlembacc.4	. . . . . . . . 9 |- ( ph -> D e. X )
13	12	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> D e. X )
14		simplr 529	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> z e. D )
15		tfr1onlembacc.u	. . . . . . . . . 10 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
16	15	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. U. X ) -> suc x e. X )
17	16	adantlr 477	. . . . . . . 8 |- ( ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) /\ x e. U. X ) -> suc x e. X )
18		simpr1 1030	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g Fn z )
19		simpr2 1031	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g e. A )
20	3, 5, 7, 10, 11, 13, 14, 17, 18, 19	tfr1onlemsucaccv 6572	. . . . . . 7 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> ( g u. { <. z , ( G ` g ) >. } ) e. A )
21	2, 20	eqeltrd 2309	. . . . . 6 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> h e. A )
22	21	ex 115	. . . . 5 |- ( ( ph /\ z e. D ) -> ( ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
23	22	exlimdv 1868	. . . 4 |- ( ( ph /\ z e. D ) -> ( E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
24	23	rexlimdva 2660	. . 3 |- ( ph -> ( E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
25	24	abssdv 3312	. 2 |- ( ph -> { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) } C_ A )
26	1, 25	eqsstrid 3284	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2813   u. cun 3209   C_ wss 3211   {csn 3689   <.cop 3692   U.cuni 3914   Ord word 4483   suc csuc 4486   |` cres 4751   Fun wfun 5346   Fn wfn 5347   ` cfv 5352  recscrecs 6535

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  tfr1onlembfn  6575  tfr1onlemubacc  6577