Theorem tfr1onlemssrecs 6166

Description: Lemma for tfr1on 6177. The union of functions acceptable for tfr1on 6177 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 4346	. . . . . 6 |- ( Ord X -> X C_ On )
4		ssrexv 3109	. . . . . 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 3117	. . . 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	syl5eqss 3093	. . 3 |- ( ph -> A C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
8	7	unissd 3707	. 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 6132	. 2 |- recs ( G ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
10	8, 9	syl6sseqr 3096	1 |- ( ph -> U. A C_ recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   {cab 2086   A.wral 2375   E.wrex 2376   C_ wss 3021   U.cuni 3683   Ord word 4222   Oncon0 4223   |` cres 4479   Fn wfn 5054   ` cfv 5059  recscrecs 6131

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-recs 6132

This theorem is referenced by:  tfr1onlembfn  6171  tfr1onlemubacc  6173  tfr1onlemres  6176