Theorem tfr1onlembacc 6283

Description: Lemma for tfr1on 6291. 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 990	. . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 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 1213	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
10	9	3adant1r 1213	. . . . . . . 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 480	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> D e. X )
14		simplr 520	. . . . . . . 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 469	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. U. X ) -> suc x e. X )
17	16	adantlr 469	. . . . . . . 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 988	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g Fn z )
19		simpr2 989	. . . . . . . 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 6282	. . . . . . 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 2234	. . . . . 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 1799	. . . 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 2574	. . 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 3202	. 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 3174	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   E.wex 1472   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   _Vcvv 2712   u. cun 3100   C_ wss 3102   {csn 3560   <.cop 3563   U.cuni 3772   Ord word 4321   suc csuc 4324   |` cres 4585   Fun wfun 5161   Fn wfn 5162   ` cfv 5167  recscrecs 6245

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-res 4595  df-iota 5132  df-fun 5169  df-fn 5170  df-fv 5175

This theorem is referenced by:  tfr1onlembfn  6285  tfr1onlemubacc  6287