Theorem unceq 37103

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 6901	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
2	1	breqd 5163	. . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧))
3	2	oprabbidv 7492	. 2 ⊢ (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧})
4		df-unc 8280	. 2 ⊢ uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧}
5		df-unc 8280	. 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 1533   class class class wbr 5152  ‘cfv 6553  {coprab 7427  uncurry cunc 8278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-oprab 7430  df-unc 8280

This theorem is referenced by: (None)

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