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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.
StepHypRef Expression
1 dmeq 5914 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
21dmeqd 5916 . . 3 (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3 breq 5145 . . . 4 (𝐴 = 𝐵 → (⟨𝑥, 𝑦𝐴𝑧 ↔ ⟨𝑥, 𝑦𝐵𝑧))
43opabbidv 5209 . . 3 (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
52, 4mpteq12dv 5233 . 2 (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧}))
6 df-cur 8292 . 2 curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧})
7 df-cur 8292 . 2 curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
85, 6, 73eqtr4g 2802 1 (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Colors of variables: wff setvar class
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
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