Theorem rdgeq2 8405

Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
rdgeq2	⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))

Proof of Theorem rdgeq2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ifeq1 4493	. . . 4 ⊢ (𝐴 = 𝐵 → if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
2	1	mpteq2dv 5207	. . 3 ⊢ (𝐴 = 𝐵 → (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
3		recseq 8366	. . 3 ⊢ ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))))
4	2, 3	syl 18	. 2 ⊢ (𝐴 = 𝐵 → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))))
5		df-rdg 8403	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
6		df-rdg 8403	. 2 ⊢ rec(𝐹, 𝐵) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐵, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
7	4, 5, 6	3eqtr4g 2825	1 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3457  ∅c0 4286  ifcif 4489  ∪ cuni 4874   ↦ cmpt 5194  dom cdm 5663  ran crn 5664  Lim wlim 6365  ‘cfv 6540  recscrecs 8363  reccrdg 8402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fv 6548  df-ov 7422  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by:  rdgeq12  8406  rdg0g  8420  oav  8502  itunifval  10415  hsmex  10431  ltweuz  14015  seqeq1  14058  dfrdg2  36322  ttcid  37060  finxpeq2  38090  finxpreclem6  38099  finxpsuclem  38100