Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cureq Structured version   Visualization version   GIF version

Theorem cureq 34004
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 5569 . . . 4 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
21dmeqd 5571 . . 3 (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3 breq 4888 . . . 4 (𝐴 = 𝐵 → (⟨𝑥, 𝑦𝐴𝑧 ↔ ⟨𝑥, 𝑦𝐵𝑧))
43opabbidv 4952 . . 3 (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
52, 4mpteq12dv 4969 . 2 (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧}))
6 df-cur 7675 . 2 curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐴𝑧})
7 df-cur 7675 . 2 curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐵𝑧})
85, 6, 73eqtr4g 2838 1 (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  cop 4403   class class class wbr 4886  {copab 4948  cmpt 4965  dom cdm 5355  curry ccur 7673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-mpt 4966  df-dm 5365  df-cur 7675
This theorem is referenced by:  curfv  34008  matunitlindf  34027
  Copyright terms: Public domain W3C validator