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Theorem mpodifsnif 7526
Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
Assertion
Ref Expression
mpodifsnif (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵𝐷)

Proof of Theorem mpodifsnif
StepHypRef Expression
1 eldifsnneq 4795 . . . 4 (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
21adantr 480 . . 3 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → ¬ 𝑖 = 𝑋)
32iffalsed 4540 . 2 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
43mpoeq3ia 7490 1 (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2105  cdif 3946  ifcif 4529  {csn 4629  cmpo 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3952  df-if 4530  df-sn 4630  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  smadiadetglem1  22394
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