Theorem rdgeq1 8422

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 6884	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐺‘(𝑔‘∪ dom 𝑔)))
2	1	ifeq2d 4502	. . . . 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 4502	. . . 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 5198	. . 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 8384	. . 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 18	. 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 8421	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
8		df-rdg 8421	. 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 2820	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3450  ∅c0 4278  ifcif 4481  ∪ cuni 4866   ↦ cmpt 5185  dom cdm 5650  ran crn 5651  Lim wlim 6363  ‘cfv 6538  recscrecs 8381  reccrdg 8420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5656  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7423  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421

This theorem is used by:  rdgeq12  8424  rdgsucmpt2  8441  frsucmpt2  8451  seqomlem0  8462  omv  8523  oev  8525  dffi3  9426  hsmex  10513  axdc  10602  seqeq2  14151  seqval  14158  precsexlemcbv  28597  seqsval  28679  seqsfn  28700  seqsp1  28702  constrcbvlem  34433  neibastop2  37241  rdgssun  38394  exrecfnlem  38395  dffinxpf  38401  finxpeq1  38402