Theorem rdgeq1 6426

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 variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5554	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘𝑥)) = (𝐺‘(𝑔‘𝑥)))
2	1	iuneq2d 3938	. . . . 5 ⊢ (𝐹 = 𝐺 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))
3	2	uneq2d 3314	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))
4	3	mpteq2dv 4121	. . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
5		recseq 6361	. . 3 ⊢ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))) → recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))))
6	4, 5	syl 14	. 2 ⊢ (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))))
7		df-irdg 6425	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
8		df-irdg 6425	. 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
9	6, 7, 8	3eqtr4g 2251	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2760   ∪ cun 3152  ∪ ciun 3913   ↦ cmpt 4091  dom cdm 4660  ‘cfv 5255  recscrecs 6359  reccrdg 6424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-iota 5216  df-fv 5263  df-recs 6360  df-irdg 6425

This theorem is referenced by:  omv  6510  oeiv  6511