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Theorem tfrlem6 8005
 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 5659 . . 3 (Rel 𝐴 ↔ ∀𝑔𝐴 Rel 𝑔)
2 tfrlem.1 . . . . 5 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
32tfrlem4 8002 . . . 4 (𝑔𝐴 → Fun 𝑔)
4 funrel 6345 . . . 4 (Fun 𝑔 → Rel 𝑔)
53, 4syl 17 . . 3 (𝑔𝐴 → Rel 𝑔)
61, 5mprgbir 3124 . 2 Rel 𝐴
72recsfval 8004 . . 3 recs(𝐹) = 𝐴
87releqi 5620 . 2 (Rel recs(𝐹) ↔ Rel 𝐴)
96, 8mpbir 234 1 Rel recs(𝐹)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {cab 2779  ∀wral 3109  ∃wrex 3110  ∪ cuni 4803   ↾ cres 5525  Rel wrel 5528  Oncon0 6163  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328  recscrecs 7994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-wrecs 7934  df-recs 7995 This theorem is referenced by:  tfrlem7  8006  tfrlem11  8011  tfrlem15  8015  tfrlem16  8016
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