Theorem tfr1onlemssrecs 6342

Description: Lemma for tfr1on 6353. The union of functions acceptable for tfr1on 6353 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 4493	. . . . . 6 |- ( Ord X -> X C_ On )
4		ssrexv 3222	. . . . . 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 3230	. . . 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 3203	. . 3 |- ( ph -> A C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
8	7	unissd 3835	. 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 6308	. 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 3206	1 |- ( ph -> U. A C_ recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   {cab 2163   A.wral 2455   E.wrex 2456   C_ wss 3131   U.cuni 3811   Ord word 4364   Oncon0 4365   |` cres 4630   Fn wfn 5213   ` cfv 5218  recscrecs 6307

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-recs 6308

This theorem is referenced by:  tfr1onlembfn  6347  tfr1onlemubacc  6349  tfr1onlemres  6352