Theorem rdgeq1 6362

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 5506	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘𝑥)) = (𝐺‘(𝑔‘𝑥)))
2	1	iuneq2d 3907	. . . . 5 ⊢ (𝐹 = 𝐺 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))
3	2	uneq2d 3287	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))
4	3	mpteq2dv 4089	. . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
5		recseq 6297	. . 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 6361	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
8		df-irdg 6361	. 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
9	6, 7, 8	3eqtr4g 2233	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1353  Vcvv 2735   ∪ cun 3125  ∪ ciun 3882   ↦ cmpt 4059  dom cdm 4620  ‘cfv 5208  recscrecs 6295  reccrdg 6360

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-iota 5170  df-fv 5216  df-recs 6296  df-irdg 6361

This theorem is referenced by:  omv  6446  oeiv  6447