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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
tfr1onlemssrecs.x
Assertion
Ref Expression
tfr1onlemssrecs recs
Distinct variable groups:   ,,,   ,   ,
Allowed substitution hints:   (,)   (,,)   (,)

Proof of Theorem tfr1onlemssrecs
StepHypRef Expression
1 tfr1onlemssrecs.1 . . . 4
2 tfr1onlemssrecs.x . . . . . 6
3 ordsson 4346 . . . . . 6
4 ssrexv 3109 . . . . . 6
52, 3, 43syl 17 . . . . 5
65ss2abdv 3117 . . . 4
71, 6syl5eqss 3093 . . 3
87unissd 3707 . 2
9 df-recs 6132 . 2 recs
108, 9syl6sseqr 3096 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1299  cab 2086  wral 2375  wrex 2376   wss 3021  cuni 3683   word 4222  con0 4223   cres 4479   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
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