Theorem rdgeq1 6515

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 5625	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘𝑥)) = (𝐺‘(𝑔‘𝑥)))
2	1	iuneq2d 3989	. . . . 5 ⊢ (𝐹 = 𝐺 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))
3	2	uneq2d 3358	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))
4	3	mpteq2dv 4174	. . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
5		recseq 6450	. . 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 6514	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
8		df-irdg 6514	. 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
9	6, 7, 8	3eqtr4g 2287	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2799   ∪ cun 3195  ∪ ciun 3964   ↦ cmpt 4144  dom cdm 4718  ‘cfv 5317  recscrecs 6448  reccrdg 6513

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-iota 5277  df-fv 5325  df-recs 6449  df-irdg 6514

This theorem is referenced by:  omv  6599  oeiv  6600