Theorem unceq 37798

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 6833	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	breqd 5109	. . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧))
3	2	oprabbidv 7424	. 2 ⊢ (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧})
4		df-unc 8210	. 2 ⊢ uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧}
5		df-unc 8210	. 2 ⊢ uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧}
6	3, 4, 5	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Syntax hints:   → wi 4   = wceq 1541   class class class wbr 5098  ‘cfv 6492  {coprab 7359  uncurry cunc 8208

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 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-ss 3918  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-oprab 7362  df-unc 8210

This theorem is referenced by: (None)