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Theorem mpodifsnif 7284
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 4680 . . . 4 (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
21adantr 484 . . 3 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → ¬ 𝑖 = 𝑋)
32iffalsed 4426 . 2 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
43mpoeq3ia 7249 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 399   = wceq 1542  wcel 2114  cdif 3841  ifcif 4415  {csn 4517  cmpo 7175
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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-v 3401  df-dif 3847  df-if 4416  df-sn 4518  df-oprab 7177  df-mpo 7178
This theorem is referenced by:  smadiadetglem1  21425
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