Theorem mposnif 7533

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 4604	. . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
2	1	adantr 486	. . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋)
3	2	iftrued 4493	. 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
4	3	mpoeq3ia 7495	1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ifcif 4485  {csn 4587   ∈ cmpo 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-if 4486  df-sn 4588  df-oprab 7421  df-mpo 7422

This theorem is used by:  mdetrsca2  22832  mdetrlin2  22835  mdetunilem5  22844  smadiadetglem2  22900