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Theorem tfrlem6 5932
 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 4460 . . 3 (Rel 𝐴 ↔ ∀𝑔𝐴 Rel 𝑔)
2 tfrlem.1 . . . . 5 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
32tfrlem4 5929 . . . 4 (𝑔𝐴 → Fun 𝑔)
4 funrel 4919 . . . 4 (Fun 𝑔 → Rel 𝑔)
53, 4syl 14 . . 3 (𝑔𝐴 → Rel 𝑔)
61, 5mprgbir 2379 . 2 Rel 𝐴
72recsfval 5931 . . 3 recs(𝐹) = 𝐴
87releqi 4423 . 2 (Rel recs(𝐹) ↔ Rel 𝐴)
96, 8mpbir 134 1 Rel recs(𝐹)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  ∪ cuni 3580  Oncon0 4100   ↾ cres 4347  Rel wrel 4350  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902  recscrecs 5919 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920 This theorem is referenced by:  tfrlem7  5933
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