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Theorem cureq 37583
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 5917 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
21dmeqd 5919 . . 3 (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3 breq 5150 . . . 4 (𝐴 = 𝐵 → (⟨𝑥, 𝑦𝐴𝑧 ↔ ⟨𝑥, 𝑦𝐵𝑧))
43opabbidv 5214 . . 3 (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
52, 4mpteq12dv 5239 . 2 (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧}))
6 df-cur 8291 . 2 curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧})
7 df-cur 8291 . 2 curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
85, 6, 73eqtr4g 2800 1 (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cop 4637   class class class wbr 5148  {copab 5210  cmpt 5231  dom cdm 5689  curry ccur 8289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-dm 5699  df-cur 8291
This theorem is referenced by:  curfv  37587  matunitlindf  37605
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