Theorem tfr1onlemssrecs 6397

Description: Lemma for tfr1on 6408. The union of functions acceptable for tfr1on 6408 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 4528	. . . . . 6 |- ( Ord X -> X C_ On )
4		ssrexv 3248	. . . . . 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 3256	. . . 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 3229	. . 3 |- ( ph -> A C_ { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } )
8	7	unissd 3863	. 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 6363	. 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 3232	1 |- ( ph -> U. A C_ recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   {cab 2182   A.wral 2475   E.wrex 2476   C_ wss 3157   U.cuni 3839   Ord word 4397   Oncon0 4398   |` cres 4665   Fn wfn 5253   ` cfv 5258  recscrecs 6362

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-recs 6363

This theorem is referenced by:  tfr1onlembfn  6402  tfr1onlemubacc  6404  tfr1onlemres  6407