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Theorem mposnif 7549
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}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)

Proof of Theorem mposnif
StepHypRef Expression
1 elsni 4648 . . . 4 (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
21adantr 480 . . 3 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → 𝑖 = 𝑋)
32iftrued 4539 . 2 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
43mpoeq3ia 7511 1 (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  ifcif 4531  {csn 4631  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532  df-sn 4632  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  mdetrsca2  22626  mdetrlin2  22629  mdetunilem5  22638  smadiadetglem2  22694
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