Theorem tfr1onlembacc 6321

Description: Lemma for tfr1on 6329. 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 1000	. . . . . . 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 485	. . . . . . . 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 485	. . . . . . . 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 1226	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
10	9	3adant1r 1226	. . . . . . . 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 485	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> D e. X )
14		simplr 525	. . . . . . . 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 474	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. U. X ) -> suc x e. X )
17	16	adantlr 474	. . . . . . . 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 998	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g Fn z )
19		simpr2 999	. . . . . . . 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 6320	. . . . . . 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 2247	. . . . . 6 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> h e. A )
22	21	ex 114	. . . . 5 |- ( ( ph /\ z e. D ) -> ( ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
23	22	exlimdv 1812	. . . 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 2587	. . 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 3221	. 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 3193	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730   u. cun 3119   C_ wss 3121   {csn 3583   <.cop 3586   U.cuni 3796   Ord word 4347   suc csuc 4350   |` cres 4613   Fun wfun 5192   Fn wfn 5193   ` cfv 5198  recscrecs 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tfr1onlembfn  6323  tfr1onlemubacc  6325