Theorem tfr1onlembacc 6395

Description: Lemma for tfr1on 6403. 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 1007	. . . . . . 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 1233	. . . . . . . . 9 |- ( ( ( ph /\ z e. D ) /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
10	9	3adant1r 1233	. . . . . . . 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 528	. . . . . . . 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 1005	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g Fn z )
19		simpr2 1006	. . . . . . . 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 6394	. . . . . . 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 2270	. . . . . 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 1830	. . . 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 2611	. . 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 3253	. 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 3225	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   u. cun 3151   C_ wss 3153   {csn 3618   <.cop 3621   U.cuni 3835   Ord word 4393   suc csuc 4396   |` cres 4661   Fun wfun 5248   Fn wfn 5249   ` cfv 5254  recscrecs 6357

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  tfr1onlembfn  6397  tfr1onlemubacc  6399