MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdgeq1 Structured version   Visualization version   GIF version

Theorem rdgeq1 8358
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 variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6842 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(𝑔 dom 𝑔)) = (𝐺‘(𝑔 dom 𝑔)))
21ifeq2d 4507 . . . . 5 (𝐹 = 𝐺 → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))
32ifeq2d 4507 . . . 4 (𝐹 = 𝐺 → if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))
43mpteq2dv 5208 . . 3 (𝐹 = 𝐺 → (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
5 recseq 8321 . . 3 ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))) → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))))
64, 5syl 17 . 2 (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))))
7 df-rdg 8357 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
8 df-rdg 8357 . 2 rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
96, 7, 83eqtr4g 2802 1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3446  c0 4283  ifcif 4487   cuni 4866  cmpt 5189  dom cdm 5634  ran crn 5635  Lim wlim 6319  cfv 6497  recscrecs 8317  reccrdg 8356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-xp 5640  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fv 6505  df-ov 7361  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357
This theorem is referenced by:  rdgeq12  8360  rdgsucmpt2  8377  frsucmpt2  8387  seqomlem0  8396  omv  8459  oev  8461  dffi3  9368  hsmex  10369  axdc  10458  seqeq2  13911  seqval  13918  neibastop2  34836  rdgssun  35852  exrecfnlem  35853  dffinxpf  35859  finxpeq1  35860
  Copyright terms: Public domain W3C validator