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Theorem tfr2a 5936
 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 2040 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem9 5935 . . 3 (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
3 tfr.1 . . . 4 𝐹 = recs(𝐺)
43dmeqi 4536 . . 3 dom 𝐹 = dom recs(𝐺)
52, 4eleq2s 2132 . 2 (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴)))
63fveq1i 5179 . 2 (𝐹𝐴) = (recs(𝐺)‘𝐴)
73reseq1i 4608 . . 3 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87fveq2i 5181 . 2 (𝐺‘(𝐹𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴))
95, 6, 83eqtr4g 2097 1 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Oncon0 4100  dom cdm 4345   ↾ cres 4347   Fn wfn 4897  ‘cfv 4902  recscrecs 5919 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920 This theorem is referenced by:  tfr0  5937  tfri2d  5950  tfri2  5952
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