Theorem cureq 37603

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 5914	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
2	1	dmeqd 5916	. . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3		breq 5145	. . . 4 ⊢ (𝐴 = 𝐵 → (⟨𝑥, 𝑦⟩𝐴𝑧 ↔ ⟨𝑥, 𝑦⟩𝐵𝑧))
4	3	opabbidv 5209	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
5	2, 4	mpteq12dv 5233	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}))
6		df-cur 8292	. 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧})
7		df-cur 8292	. 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
8	5, 6, 7	3eqtr4g 2802	1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

Syntax hints:   → wi 4   = wceq 1540  ⟨cop 4632   class class class wbr 5143  {copab 5205   ↦ cmpt 5225  dom cdm 5685  curry ccur 8290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-dm 5695  df-cur 8292

This theorem is referenced by:  curfv  37607  matunitlindf  37625