Theorem rdgeq1 6374

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 5516	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑔‘𝑥)) = (𝐺‘(𝑔‘𝑥)))
2	1	iuneq2d 3913	. . . . 5 ⊢ (𝐹 = 𝐺 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))
3	2	uneq2d 3291	. . . 4 ⊢ (𝐹 = 𝐺 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥))))
4	3	mpteq2dv 4096	. . 3 ⊢ (𝐹 = 𝐺 → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
5		recseq 6309	. . 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 6373	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
8		df-irdg 6373	. 2 ⊢ rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐺‘(𝑔‘𝑥)))))
9	6, 7, 8	3eqtr4g 2235	1 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Syntax hints:   → wi 4   = wceq 1353  Vcvv 2739   ∪ cun 3129  ∪ ciun 3888   ↦ cmpt 4066  dom cdm 4628  ‘cfv 5218  recscrecs 6307  reccrdg 6372

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-iota 5180  df-fv 5226  df-recs 6308  df-irdg 6373

This theorem is referenced by:  omv  6458  oeiv  6459