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Theorem tfr2a 5964
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 2054 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem9 5963 . . 3 (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 4561 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2146 . 2 (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
63fveq1i 5204 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
73reseq1i 4633 . . 3 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87fveq2i 5206 . 2 (𝐺‘(𝐹𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
95, 6, 83eqtr4g 2111 1 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wcel 1407  {cab 2040  wral 2321  wrex 2322  Oncon0 4125  dom cdm 4370  cres 4372   Fn wfn 4922  cfv 4927  recscrecs 5947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-setind 4287
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-res 4382  df-iota 4892  df-fun 4929  df-fn 4930  df-fv 4935  df-recs 5948
This theorem is referenced by:  tfr0  5965  tfri2d  5978  tfri2  5980
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