Theorem cureq 36967

Description: Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
cureq	⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

Proof of Theorem cureq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5894	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
2	1	dmeqd 5896	. . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3		breq 5141	. . . 4 ⊢ (𝐴 = 𝐵 → (⟨𝑥, 𝑦⟩𝐴𝑧 ↔ ⟨𝑥, 𝑦⟩𝐵𝑧))
4	3	opabbidv 5205	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
5	2, 4	mpteq12dv 5230	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}))
6		df-cur 8248	. 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧})
7		df-cur 8248	. 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
8	5, 6, 7	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

Syntax hints:   → wi 4   = wceq 1533  ⟨cop 4627   class class class wbr 5139  {copab 5201   ↦ cmpt 5222  dom cdm 5667  curry ccur 8246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-mpt 5223  df-dm 5677  df-cur 8248

This theorem is referenced by:  curfv  36971  matunitlindf  36989