MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2difsnif Structured version   Visualization version   GIF version

Theorem mpt2difsnif 6750
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 4315 . . . . 5 (𝑖 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑖𝐴𝑖𝑋))
2 neneq 2799 . . . . 5 (𝑖𝑋 → ¬ 𝑖 = 𝑋)
31, 2simplbiim 659 . . . 4 (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
43adantr 481 . . 3 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → ¬ 𝑖 = 𝑋)
54iffalsed 4095 . 2 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
65mpt2eq3ia 6717 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 384   = wceq 1482  wcel 1989  wne 2793  cdif 3569  ifcif 4084  {csn 4175  cmpt2 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-if 4085  df-sn 4176  df-oprab 6651  df-mpt2 6652
This theorem is referenced by:  smadiadetglem1  20471
  Copyright terms: Public domain W3C validator