Theorem tfrcllembacc 6370

Description: Lemma for tfrcl 6379. Each element of B is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem tfrcllembacc

Step	Hyp	Ref	Expression
1		tfrcllembacc.3	. 2 |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
2		simpr3 1006	. . . . . . 7 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> h = ( g u. { <. z , ( G ` g ) >. } ) )
3		tfrcl.f	. . . . . . . 8 |- F = recs ( G )
4		tfrcl.g	. . . . . . . . 9 |- ( ph -> Fun G )
5	4	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> Fun G )
6		tfrcl.x	. . . . . . . . 9 |- ( ph -> Ord X )
7	6	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> Ord X )
8		simp1ll 1061	. . . . . . . . 9 |- ( ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) /\ x e. X /\ f : x --> S ) -> ph )
9		tfrcl.ex	. . . . . . . . 9 |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )
10	8, 9	syld3an1 1294	. . . . . . . 8 |- ( ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )
11		tfrcllemsucfn.1	. . . . . . . 8 |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
12		tfrcllembacc.4	. . . . . . . . 9 |- ( ph -> D e. X )
13	12	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> D e. X )
14		simplr 528	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> z e. D )
15		tfrcllembacc.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 : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) /\ x e. U. X ) -> suc x e. X )
18		simpr1 1004	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g : z --> S )
19		simpr2 1005	. . . . . . . 8 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> g e. A )
20	3, 5, 7, 10, 11, 13, 14, 17, 18, 19	tfrcllemsucaccv 6369	. . . . . . 7 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> ( g u. { <. z , ( G ` g ) >. } ) e. A )
21	2, 20	eqeltrd 2264	. . . . . 6 |- ( ( ( ph /\ z e. D ) /\ ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) ) -> h e. A )
22	21	ex 115	. . . . 5 |- ( ( ph /\ z e. D ) -> ( ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
23	22	exlimdv 1829	. . . 4 |- ( ( ph /\ z e. D ) -> ( E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
24	23	rexlimdva 2604	. . 3 |- ( ph -> ( E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) -> h e. A ) )
25	24	abssdv 3241	. 2 |- ( ph -> { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) } C_ A )
26	1, 25	eqsstrid 3213	1 |- ( ph -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 979   = wceq 1363   E.wex 1502   e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   u. cun 3139   C_ wss 3141   {csn 3604   <.cop 3607   U.cuni 3821   Ord word 4374   suc csuc 4377   |` cres 4640   Fun wfun 5222   -->wf 5224   ` cfv 5228  recscrecs 6319

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236

This theorem is referenced by:  tfrcllembfn  6372  tfrcllemubacc  6374