Theorem unceq 37070

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 6896	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	breqd 5159	. . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧))
3	2	oprabbidv 7486	. 2 ⊢ (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧})
4		df-unc 8273	. 2 ⊢ uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧}
5		df-unc 8273	. 2 ⊢ uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧}
6	3, 4, 5	3eqtr4g 2793	1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Syntax hints:   → wi 4   = wceq 1534   class class class wbr 5148  ‘cfv 6548  {coprab 7421  uncurry cunc 8271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-oprab 7424  df-unc 8273

This theorem is referenced by: (None)