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Theorem tfrlem6 5873
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypothesis
Ref Expression
tfrlem.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem6 Rel recs(𝐹)
Distinct variable group:   x,f,y,𝐹
Allowed substitution hints:   A(x,y,f)

Proof of Theorem tfrlem6
Dummy variable g is distinct from all other variables.
StepHypRef Expression
1 reluni 4403 . . 3 (Rel Ag A Rel g)
2 tfrlem.1 . . . . 5 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
32tfrlem4 5870 . . . 4 (g A → Fun g)
4 funrel 4862 . . . 4 (Fun g → Rel g)
53, 4syl 14 . . 3 (g A → Rel g)
61, 5mprgbir 2373 . 2 Rel A
72recsfval 5872 . . 3 recs(𝐹) = A
87releqi 4366 . 2 (Rel recs(𝐹) ↔ Rel A)
96, 8mpbir 134 1 Rel recs(𝐹)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301   cuni 3571  Oncon0 4066  cres 4290  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  cfv 4845  recscrecs 5860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfrlem7  5874
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