Theorem cureq 8889

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.

Step	Hyp	Ref	Expression
1		dmeq 5884	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
2	1	dmeqd 5886	. . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3		breq 5104	. . . 4 ⊢ (𝐴 = 𝐵 → (⟨𝑥, 𝑦⟩𝐴𝑧 ↔ ⟨𝑥, 𝑦⟩𝐵𝑧))
4	3	opabbidv 5170	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
5	2, 4	mpteq12dv 5191	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}))
6		df-cur 8284	. 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧})
7		df-cur 8284	. 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
8	5, 6, 7	3eqtr4g 2820	1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⟨cop 4589   class class class wbr 5102  {copab 5166   ↦ cmpt 5185  dom cdm 5650  curry ccur 8282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-dm 5660  df-cur 8284

This theorem is used by:  curfv  8892  matunitlindf  22996