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Theorem tfr2a 5858
Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2a (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))

Proof of Theorem tfr2a
Dummy variables x f y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . 4 {fx On (f Fn x y x (fy) = (𝐺‘(fy)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
21tfrlem9 5857 . . 3 (A dom recs(𝐺) → (recs(𝐺)‘A) = (𝐺‘(recs(𝐺) ↾ A)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 4463 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2114 . 2 (A dom 𝐹 → (recs(𝐺)‘A) = (𝐺‘(recs(𝐺) ↾ A)))
63fveq1i 5104 . 2 (𝐹A) = (recs(𝐺)‘A)
73reseq1i 4535 . . 3 (𝐹A) = (recs(𝐺) ↾ A)
87fveq2i 5106 . 2 (𝐺‘(𝐹A)) = (𝐺‘(recs(𝐺) ↾ A))
95, 6, 83eqtr4g 2079 1 (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  {cab 2008  wral 2284  wrex 2285  Oncon0 4049  dom cdm 4272  cres 4274   Fn wfn 4824  cfv 4829  recscrecs 5841
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-recs 5842
This theorem is referenced by:  tfri2d  5872  tfri2  5874
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