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Theorem rdgeq2 6237
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 3193 . . . 4 (𝐴 = 𝐵 → (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))) = (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))
21mpteq2dv 3989 . . 3 (𝐴 = 𝐵 → (𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))) = (𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
3 recseq 6171 . . 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 6235 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
6 df-irdg 6235 . 2 rec(𝐹, 𝐵) = recs((𝑔 ∈ V ↦ (𝐵 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
74, 5, 63eqtr4g 2175 1 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  Vcvv 2660  cun 3039   ciun 3783  cmpt 3959  dom cdm 4509  cfv 5093  recscrecs 6169  reccrdg 6234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-iota 5058  df-fv 5101  df-recs 6170  df-irdg 6235
This theorem is referenced by:  rdg0g  6253  oav  6318
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