Theorem rdgeq1 8213

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 6755	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐺‘(𝑔‘∪ dom 𝑔)))
2	1	ifeq2d 4476	. . . . 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 4476	. . . 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 5172	. . 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 8176	. . 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 8212	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
8		df-rdg 8212	. 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 2804	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1539  Vcvv 3422  ∅c0 4253  ifcif 4456  ∪ cuni 4836   ↦ cmpt 5153  dom cdm 5580  ran crn 5581  Lim wlim 6252  ‘cfv 6418  recscrecs 8172  reccrdg 8211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fv 6426  df-ov 7258  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212

This theorem is referenced by:  rdgeq12  8215  rdgsucmpt2  8232  frsucmpt2  8241  seqomlem0  8250  omv  8304  oev  8306  dffi3  9120  trpredlem1  9405  trpredtr  9408  trpredmintr  9409  hsmex  10119  axdc  10208  seqeq2  13653  seqval  13660  neibastop2  34477  rdgssun  35476  exrecfnlem  35477  dffinxpf  35483  finxpeq1  35484