Theorem tfr1onlemssrecs 6318

Description: Lemma for tfr1on 6329. The union of functions acceptable for tfr1on 6329 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)

Hypotheses

Ref	Expression
tfr1onlemssrecs.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfr1onlemssrecs.x	|- ( ph -> Ord X )

Assertion

Ref	Expression
tfr1onlemssrecs	|- ( ph -> U. A C_ recs ( G ) )

Distinct variable groups:   f, G, x, y   x, X   ph, f

Allowed substitution hints:   ph( x, y)   A( x, y, f)   X( y, f)

Proof of Theorem tfr1onlemssrecs

Step	Hyp	Ref	Expression
1		tfr1onlemssrecs.1	. . . 4 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
2		tfr1onlemssrecs.x	. . . . . 6 |- ( ph -> Ord X )
3		ordsson 4476	. . . . . 6 |- ( Ord X -> X C_ On )
4		ssrexv 3212	. . . . . 6 |- ( X C_ On -> ( E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) -> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) )
5	2, 3, 4	3syl 17	. . . . 5 |- ( ph -> ( E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) -> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) ) )
6	5	ss2abdv 3220	. . . 4 |- ( ph -> { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
7	1, 6	eqsstrid 3193	. . 3 |- ( ph -> A C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
8	7	unissd 3820	. 2 |- ( ph -> U. A C_ U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
9		df-recs 6284	. 2 |- recs ( G ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
10	8, 9	sseqtrrdi 3196	1 |- ( ph -> U. A C_ recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   {cab 2156   A.wral 2448   E.wrex 2449   C_ wss 3121   U.cuni 3796   Ord word 4347   Oncon0 4348   |` cres 4613   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-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-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-recs 6284

This theorem is referenced by:  tfr1onlembfn  6323  tfr1onlemubacc  6325  tfr1onlemres  6328