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Theorem frrlem6 32066
 Description: Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Scott Fenton, 23-Dec-2021.)
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 = 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 reluni 5385 . . 3 (Rel 𝐵 ↔ ∀𝑔𝐵 Rel 𝑔)
2 frrlem6.1 . . . . 5 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
32frrlem2 32058 . . . 4 (𝑔𝐵 → Fun 𝑔)
4 funrel 6054 . . . 4 (Fun 𝑔 → Rel 𝑔)
53, 4syl 17 . . 3 (𝑔𝐵 → Rel 𝑔)
61, 5mprgbir 3053 . 2 Rel 𝐵
7 frrlem6.2 . . . . 5 𝐹 = frecs(𝑅, 𝐴, 𝐺)
8 df-frecs 32053 . . . . 5 frecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
97, 8eqtri 2770 . . . 4 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
102unieqi 4585 . . . 4 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
119, 10eqtr4i 2773 . . 3 𝐹 = 𝐵
1211releqi 5347 . 2 (Rel 𝐹 ↔ Rel 𝐵)
136, 12mpbir 221 1 Rel 𝐹
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∧ w3a 1072   = wceq 1620  ∃wex 1841   ∈ wcel 2127  {cab 2734  ∀wral 3038   ⊆ wss 3703  ∪ cuni 4576   ↾ cres 5256  Rel wrel 5259  Predcpred 5828  Fun wfun 6031   Fn wfn 6032  ‘cfv 6037  (class class class)co 6801  frecscfrecs 32052 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045  df-ov 6804  df-frecs 32053 This theorem is referenced by: (None)
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