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Theorem frrlem6 8278
Description: Lemma for well-founded recursion. The well-founded recursion generator is a relation. (Contributed by Scott Fenton, 27-Aug-2022.)
Hypotheses
Ref Expression
frrlem5.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem5.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
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 frrlem5.1 . . . . 5 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 frrlem5.2 . . . . 5 𝐹 = frecs(𝑅, 𝐴, 𝐺)
31, 2frrlem5 8277 . . . 4 𝐹 = 𝐵
43releqi 5777 . . 3 (Rel 𝐹 ↔ Rel 𝐵)
5 reluni 5818 . . 3 (Rel 𝐵 ↔ ∀𝑔𝐵 Rel 𝑔)
64, 5bitri 274 . 2 (Rel 𝐹 ↔ ∀𝑔𝐵 Rel 𝑔)
71frrlem2 8274 . . 3 (𝑔𝐵 → Fun 𝑔)
8 funrel 6565 . . 3 (Fun 𝑔 → Rel 𝑔)
97, 8syl 17 . 2 (𝑔𝐵 → Rel 𝑔)
106, 9mprgbir 3068 1 Rel 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wral 3061  wss 3948   cuni 4908  cres 5678  Rel wrel 5681  Predcpred 6299  Fun wfun 6537   Fn wfn 6538  cfv 6543  (class class class)co 7411  frecscfrecs 8267
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7414  df-frecs 8268
This theorem is referenced by:  frrlem9  8281  frrrel  8293
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