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Theorem tfrlem6 5965
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 4488 . . 3 (Rel 𝐴 ↔ ∀𝑔𝐴 Rel 𝑔)
2 tfrlem.1 . . . . 5 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
32tfrlem4 5962 . . . 4 (𝑔𝐴 → Fun 𝑔)
4 funrel 4949 . . . 4 (Fun 𝑔 → Rel 𝑔)
53, 4syl 14 . . 3 (𝑔𝐴 → Rel 𝑔)
61, 5mprgbir 2422 . 2 Rel 𝐴
72recsfval 5964 . . 3 recs(𝐹) = 𝐴
87releqi 4449 . 2 (Rel recs(𝐹) ↔ Rel 𝐴)
96, 8mpbir 144 1 Rel recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wcel 1434  {cab 2068  wral 2349  wrex 2350   cuni 3609  Oncon0 4126  cres 4373  Rel wrel 4376  Fun wfun 4926   Fn wfn 4927  cfv 4932  recscrecs 5953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954
This theorem is referenced by:  tfrlem7  5966
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