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Theorem cureq 34913
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 5745 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
21dmeqd 5747 . . 3 (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3 breq 5041 . . . 4 (𝐴 = 𝐵 → (⟨𝑥, 𝑦𝐴𝑧 ↔ ⟨𝑥, 𝑦𝐵𝑧))
43opabbidv 5105 . . 3 (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
52, 4mpteq12dv 5124 . 2 (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧}))
6 df-cur 7908 . 2 curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧})
7 df-cur 7908 . 2 curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
85, 6, 73eqtr4g 2881 1 (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cop 4546   class class class wbr 5039  {copab 5101  cmpt 5119  dom cdm 5528  curry ccur 7906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-mpt 5120  df-dm 5538  df-cur 7908
This theorem is referenced by:  curfv  34917  matunitlindf  34935
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