Theorem tfr1onlembex 6589

Description: Lemma for tfr1on 6594. The set B exists. (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
tfr1onlembex	|- ( ph -> B e. _V )

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   B, g, h, w, z   D, h, z   h, G, w, z, f, y, x   g, X, z

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

Proof of Theorem tfr1onlembex

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . 4 |- F = recs ( G )
2		tfr1on.g	. . . 4 |- ( ph -> Fun G )
3		tfr1on.x	. . . 4 |- ( ph -> Ord X )
4		tfr1on.ex	. . . 4 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
5		tfr1onlemsucfn.1	. . . 4 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
6		tfr1onlembacc.3	. . . 4 |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
7		tfr1onlembacc.u	. . . 4 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
8		tfr1onlembacc.4	. . . 4 |- ( ph -> D e. X )
9		tfr1onlembacc.5	. . . 4 |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembfn 6588	. . 3 |- ( ph -> U. B Fn D )
11		fnex 5911	. . 3 |- ( ( U. B Fn D /\ D e. X ) -> U. B e. _V )
12	10, 8, 11	syl2anc 411	. 2 |- ( ph -> U. B e. _V )
13		uniexb 4599	. 2 |- ( B e. _V <-> U. B e. _V )
14	12, 13	sylibr 134	1 |- ( ph -> B e. _V )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   u. cun 3212   {csn 3694   <.cop 3697   U.cuni 3919   Ord word 4488   suc csuc 4491   |` cres 4756   Fun wfun 5351   Fn wfn 5352   ` cfv 5357  recscrecs 6548

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549

This theorem is referenced by:  tfr1onlemex  6591