Theorem tfr1onlemssrecs 6425

Description: Lemma for tfr1on 6436. The union of functions acceptable for tfr1on 6436 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 4540	. . . . . 6 |- ( Ord X -> X C_ On )
4		ssrexv 3258	. . . . . 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 3266	. . . 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 3239	. . 3 |- ( ph -> A C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
8	7	unissd 3874	. 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 6391	. 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 3242	1 |- ( ph -> U. A C_ recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   {cab 2191   A.wral 2484   E.wrex 2485   C_ wss 3166   U.cuni 3850   Ord word 4409   Oncon0 4410   |` cres 4677   Fn wfn 5266   ` cfv 5271  recscrecs 6390

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-recs 6391

This theorem is referenced by:  tfr1onlembfn  6430  tfr1onlemubacc  6432  tfr1onlemres  6435