Theorem mpteq1iOLD 5166

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
mpteq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
mpteq1iOLD	⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1iOLD

Step	Hyp	Ref	Expression
1		mpteq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		mpteq1 5162	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1543   ↦ cmpt 5152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-opab 5133  df-mpt 5153

This theorem is referenced by: (None)