Theorem mposnif 7479

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 4579	. . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
2	1	adantr 481	. . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋)
3	2	iftrued 4469	. 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
4	3	mpoeq3ia 7441	1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ifcif 4461  {csn 4562   ∈ cmpo 7365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-if 4462  df-sn 4563  df-oprab 7367  df-mpo 7368

This theorem is referenced by:  mdetrsca2  22594  mdetrlin2  22597  mdetunilem5  22606  smadiadetglem2  22662