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Theorem tfrlem6 6646
 Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem6 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem6
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 reluni 5000 . . 3
2 tfrlem.1 . . . . 5
32tfrlem4 6643 . . . 4
4 funrel 5474 . . . 4
53, 4syl 16 . . 3
61, 5mprgbir 2778 . 2
72recsfval 6645 . . 3 recs
87releqi 4963 . 2 recs
96, 8mpbir 202 1 recs
 Colors of variables: wff set class Syntax hints:   wa 360   wceq 1653   wcel 1726  cab 2424  wral 2707  wrex 2708  cuni 4017  con0 4584   cres 4883   wrel 4886   wfun 5451   wfn 5452  cfv 5457  recscrecs 6635 This theorem is referenced by:  tfrlem7  6647  tfrlem11  6652  tfrlem15  6656  tfrlem16  6657 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-recs 6636
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