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Theorem frrlem5b 31029
 Description: Lemma for founded recursion. The union of a subclass of 𝐵 is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
Hypotheses
Ref Expression
frrlem5.1 𝑅 Fr 𝐴
frrlem5.2 𝑅 Se 𝐴
frrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
Assertion
Ref Expression
frrlem5b (𝐶𝐵 → Rel 𝐶)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5b
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssel 3562 . . . 4 (𝐶𝐵 → (𝑧𝐶𝑧𝐵))
2 frrlem5.3 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
32frrlem2 31025 . . . . 5 (𝑧𝐵 → Fun 𝑧)
4 funrel 5821 . . . . 5 (Fun 𝑧 → Rel 𝑧)
53, 4syl 17 . . . 4 (𝑧𝐵 → Rel 𝑧)
61, 5syl6 34 . . 3 (𝐶𝐵 → (𝑧𝐶 → Rel 𝑧))
76ralrimiv 2948 . 2 (𝐶𝐵 → ∀𝑧𝐶 Rel 𝑧)
8 reluni 5164 . 2 (Rel 𝐶 ↔ ∀𝑧𝐶 Rel 𝑧)
97, 8sylibr 223 1 (𝐶𝐵 → Rel 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372   Fr wfr 4994   Se wse 4995   ↾ cres 5040  Rel wrel 5043  Predcpred 5596  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552 This theorem is referenced by:  frrlem5c  31030
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