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Theorem mpt2difsnif 6651
 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
mpt2difsnif (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗𝐵𝐷)

Proof of Theorem mpt2difsnif
StepHypRef Expression
1 eldifsn 4260 . . . . 5 (𝑖 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑖𝐴𝑖𝑋))
2 df-ne 2782 . . . . . . 7 (𝑖𝑋 ↔ ¬ 𝑖 = 𝑋)
32biimpi 205 . . . . . 6 (𝑖𝑋 → ¬ 𝑖 = 𝑋)
43adantl 481 . . . . 5 ((𝑖𝐴𝑖𝑋) → ¬ 𝑖 = 𝑋)
51, 4sylbi 206 . . . 4 (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
65adantr 480 . . 3 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → ¬ 𝑖 = 𝑋)
76iffalsed 4047 . 2 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
87mpt2eq3ia 6618 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 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  ifcif 4036  {csn 4125   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-if 4037  df-sn 4126  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  smadiadetglem1  20296
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