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Theorem rdgeq2 5959
 Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq2 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))

Proof of Theorem rdgeq2
Dummy variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3090 . . . 4 (𝐴 = 𝐵 → (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))) = (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))
21mpteq2dv 3848 . . 3 (𝐴 = 𝐵 → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
3 recseq 5921 . . 3 ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) → recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))) = recs((𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))))
42, 3syl 14 . 2 (𝐴 = 𝐵 → recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))) = recs((𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))))
5 df-irdg 5957 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
6 df-irdg 5957 . 2 rec(𝐹, 𝐵) = recs((𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
74, 5, 63eqtr4g 2097 1 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  Vcvv 2557   ∪ cun 2915  ∪ ciun 3657   ↦ cmpt 3818  dom cdm 4345  ‘cfv 4902  recscrecs 5919  reccrdg 5956 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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-iota 4867  df-fv 4910  df-recs 5920  df-irdg 5957 This theorem is referenced by:  rdg0g  5975  oav  6034
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