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Theorem cureq 32555
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 5246 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
21dmeqd 5248 . . 3 (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3 breq 4585 . . . 4 (𝐴 = 𝐵 → (⟨𝑥, 𝑦𝐴𝑧 ↔ ⟨𝑥, 𝑦𝐵𝑧))
43opabbidv 4648 . . 3 (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
52, 4mpteq12dv 4663 . 2 (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧}))
6 df-cur 7280 . 2 curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧})
7 df-cur 7280 . 2 curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
85, 6, 73eqtr4g 2669 1 (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  cop 4131   class class class wbr 4583  {copab 4642  cmpt 4643  dom cdm 5038  curry ccur 7278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-dm 5048  df-cur 7280
This theorem is referenced by:  curfv  32559  matunitlindf  32577
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