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Theorem frrlem6 31763
Description: Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypotheses
Ref Expression
frrlem6.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
frrlem6.2 𝐹 = 𝐵
Assertion
Ref Expression
frrlem6 Rel 𝐹
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓)   𝐹(𝑥,𝑦,𝑓)

Proof of Theorem frrlem6
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 frrlem6.2 . 2 𝐹 = 𝐵
2 reluni 5231 . . . 4 (Rel 𝐵 ↔ ∀𝑔𝐵 Rel 𝑔)
3 frrlem6.1 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
43frrlem2 31755 . . . . 5 (𝑔𝐵 → Fun 𝑔)
5 funrel 5893 . . . . 5 (Fun 𝑔 → Rel 𝑔)
64, 5syl 17 . . . 4 (𝑔𝐵 → Rel 𝑔)
72, 6mprgbir 2924 . . 3 Rel 𝐵
8 releq 5191 . . 3 (𝐹 = 𝐵 → (Rel 𝐹 ↔ Rel 𝐵))
97, 8mpbiri 248 . 2 (𝐹 = 𝐵 → Rel 𝐹)
101, 9ax-mp 5 1 Rel 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wss 3567   cuni 4427  cres 5106  Rel wrel 5109  Predcpred 5667  Fun wfun 5870   Fn wfn 5871  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-ov 6638
This theorem is referenced by: (None)
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