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Theorem frrlem6 33241
Description: Lemma for founded recursion. The founded recursion generator is a relationship. (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 33240 . . . 4 𝐹 = 𝐵
43releqi 5616 . . 3 (Rel 𝐹 ↔ Rel 𝐵)
5 reluni 5655 . . 3 (Rel 𝐵 ↔ ∀𝑔𝐵 Rel 𝑔)
64, 5bitri 278 . 2 (Rel 𝐹 ↔ ∀𝑔𝐵 Rel 𝑔)
71frrlem2 33237 . . 3 (𝑔𝐵 → Fun 𝑔)
8 funrel 6341 . . 3 (Fun 𝑔 → Rel 𝑔)
97, 8syl 17 . 2 (𝑔𝐵 → Rel 𝑔)
106, 9mprgbir 3121 1 Rel 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2111  {cab 2776  wral 3106  wss 3881   cuni 4800  cres 5521  Rel wrel 5524  Predcpred 6115  Fun wfun 6318   Fn wfn 6319  cfv 6324  (class class class)co 7135  frecscfrecs 33230
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-ov 7138  df-frecs 33231
This theorem is referenced by:  frrlem9  33244
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