Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgeq2 Structured version   GIF version

Theorem rdgeq2 5899
 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 3084 . . . 4 (A = B → (A x dom g(𝐹‘(gx))) = (B x dom g(𝐹‘(gx))))
21mpteq2dv 3839 . . 3 (A = B → (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (B x dom g(𝐹‘(gx)))))
3 recseq 5862 . . 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 5897 . 2 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
6 df-irdg 5897 . 2 rec(𝐹, B) = recs((g V ↦ (B x dom g(𝐹‘(gx)))))
74, 5, 63eqtr4g 2094 1 (A = B → rec(𝐹, A) = rec(𝐹, B))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  Vcvv 2551   ∪ cun 2909  ∪ ciun 3648   ↦ cmpt 3809  dom cdm 4288  ‘cfv 4845  recscrecs 5860  reccrdg 5896 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-iota 4810  df-fv 4853  df-recs 5861  df-irdg 5897 This theorem is referenced by:  rdg0g  5915  oav  5973
 Copyright terms: Public domain W3C validator