ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgeq1 Structured version   GIF version

Theorem rdgeq1 5898
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq1 (𝐹 = 𝐺 → rec(𝐹, A) = rec(𝐺, A))

Proof of Theorem rdgeq1
Dummy variables x g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5120 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(gx)) = (𝐺‘(gx)))
21iuneq2d 3673 . . . . 5 (𝐹 = 𝐺 x dom g(𝐹‘(gx)) = x dom g(𝐺‘(gx)))
32uneq2d 3091 . . . 4 (𝐹 = 𝐺 → (A x dom g(𝐹‘(gx))) = (A x dom g(𝐺‘(gx))))
43mpteq2dv 3839 . . 3 (𝐹 = 𝐺 → (g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (A x dom g(𝐺‘(gx)))))
5 recseq 5862 . . 3 ((g V ↦ (A x dom g(𝐹‘(gx)))) = (g V ↦ (A x dom g(𝐺‘(gx)))) → recs((g V ↦ (A x dom g(𝐹‘(gx))))) = recs((g V ↦ (A x dom g(𝐺‘(gx))))))
64, 5syl 14 . 2 (𝐹 = 𝐺 → recs((g V ↦ (A x dom g(𝐹‘(gx))))) = recs((g V ↦ (A x dom g(𝐺‘(gx))))))
7 df-irdg 5897 . 2 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
8 df-irdg 5897 . 2 rec(𝐺, A) = recs((g V ↦ (A x dom g(𝐺‘(gx)))))
96, 7, 83eqtr4g 2094 1 (𝐹 = 𝐺 → rec(𝐹, A) = rec(𝐺, A))
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-bnd 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-in 2918  df-ss 2925  df-uni 3572  df-iun 3650  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:  omv  5974  oeiv  5975
  Copyright terms: Public domain W3C validator