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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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