Theorem rdgeq1 8375

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.

Step	Hyp	Ref	Expression
1		fveq1 6860	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐺‘(𝑔‘∪ dom 𝑔)))
2	1	ifeq2d 4500	. . . . 5 ⊢ (𝐹 = 𝐺 → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))
3	2	ifeq2d 4500	. . . 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)))))
4	3	mpteq2dv 5193	. . 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 8337	. . 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)))))))
6	4, 5	syl 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 8374	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
8		df-rdg 8374	. 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐺‘(𝑔‘∪ dom 𝑔))))))
9	6, 7, 8	3eqtr4g 2821	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1559  Vcvv 3453  ∅c0 4285  ifcif 4479  ∪ cuni 4864   ↦ cmpt 5180  dom cdm 5645  ran crn 5646  Lim wlim 6341  ‘cfv 6515  recscrecs 8334  reccrdg 8373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-xp 5651  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-iota 6471  df-fv 6523  df-ov 7393  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374

This theorem is referenced by:  rdgeq12  8377  rdgsucmpt2  8394  frsucmpt2  8404  seqomlem0  8413  omv  8474  oev  8476  dffi3  9372  hsmex  10384  axdc  10473  seqeq2  14013  seqval  14020  precsexlemcbv  28274  seqsval  28356  seqsfn  28377  seqsp1  28379  constrcbvlem  34011  neibastop2  36674  rdgssun  37825  exrecfnlem  37826  dffinxpf  37832  finxpeq1  37833