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Theorem tfrlem6 6310
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 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem6 Rel recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem6
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 reluni 4745 . . 3 (Rel 𝐴 ↔ ∀𝑔𝐴 Rel 𝑔)
2 tfrlem.1 . . . . 5 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
32tfrlem4 6307 . . . 4 (𝑔𝐴 → Fun 𝑔)
4 funrel 5228 . . . 4 (Fun 𝑔 → Rel 𝑔)
53, 4syl 14 . . 3 (𝑔𝐴 → Rel 𝑔)
61, 5mprgbir 2535 . 2 Rel 𝐴
72recsfval 6309 . . 3 recs(𝐹) = 𝐴
87releqi 4705 . 2 (Rel recs(𝐹) ↔ Rel 𝐴)
96, 8mpbir 146 1 Rel recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456   cuni 3807  Oncon0 4359  cres 4624  Rel wrel 4627  Fun wfun 5205   Fn wfn 5206  cfv 5211  recscrecs 6298
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-recs 6299
This theorem is referenced by:  tfrlem7  6311
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