Theorem unceq 37598

Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
unceq	⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Proof of Theorem unceq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6860	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	breqd 5121	. . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧))
3	2	oprabbidv 7458	. 2 ⊢ (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧})
4		df-unc 8250	. 2 ⊢ uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧}
5		df-unc 8250	. 2 ⊢ uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧}
6	3, 4, 5	3eqtr4g 2790	1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5110  ‘cfv 6514  {coprab 7391  uncurry cunc 8248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-oprab 7394  df-unc 8250

This theorem is referenced by: (None)