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Theorem rdgeq2 5880
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 3067 . . . 4 (A = B → (A x dom g(𝐹‘(gx))) = (B x dom g(𝐹‘(gx))))
21mpteq2dv 3822 . . 3 (A = B → (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (B x dom g(𝐹‘(gx)))))
3 recseq 5843 . . 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 5878 . 2 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
6 df-irdg 5878 . 2 rec(𝐹, B) = recs((g V ↦ (B x dom g(𝐹‘(gx)))))
74, 5, 63eqtr4g 2079 1 (A = B → rec(𝐹, A) = rec(𝐹, B))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  Vcvv 2535  cun 2892   ciun 3631  cmpt 3792  dom cdm 4272  cfv 4829  recscrecs 5841  reccrdg 5877
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-iota 4794  df-fv 4837  df-recs 5842  df-irdg 5878
This theorem is referenced by:  oav  5949
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