Theorem cureq 37635

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 5842	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
2	1	dmeqd 5844	. . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵)
3		breq 5091	. . . 4 ⊢ (𝐴 = 𝐵 → (⟨𝑥, 𝑦⟩𝐴𝑧 ↔ ⟨𝑥, 𝑦⟩𝐵𝑧))
4	3	opabbidv 5155	. . 3 ⊢ (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
5	2, 4	mpteq12dv 5176	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}))
6		df-cur 8197	. 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧})
7		df-cur 8197	. 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})
8	5, 6, 7	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

Syntax hints:   → wi 4   = wceq 1541  ⟨cop 4579   class class class wbr 5089  {copab 5151   ↦ cmpt 5170  dom cdm 5614  curry ccur 8195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-dm 5624  df-cur 8197

This theorem is referenced by:  curfv  37639  matunitlindf  37657