Theorem mposnif 7508

Description: A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)

Assertion

Ref	Expression
mposnif	⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem mposnif

Step	Hyp	Ref	Expression
1		elsni 4598	. . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
2	1	adantr 484	. . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋)
3	2	iftrued 4487	. 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
4	3	mpoeq3ia 7470	1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ifcif 4479  {csn 4581   ∈ cmpo 7394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4480  df-sn 4582  df-oprab 7396  df-mpo 7397

This theorem is referenced by:  mdetrsca2  22644  mdetrlin2  22647  mdetunilem5  22656  smadiadetglem2  22712