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

Theorem mpt2difsnif 6899
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 4451 . . . . 5 (𝑖 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑖𝐴𝑖𝑋))
2 neneq 2948 . . . . 5 (𝑖𝑋 → ¬ 𝑖 = 𝑋)
31, 2simplbiim 487 . . . 4 (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
43adantr 466 . . 3 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → ¬ 𝑖 = 𝑋)
54iffalsed 4234 . 2 ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
65mpt2eq3ia 6866 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 382   = wceq 1630  wcel 2144  wne 2942  cdif 3718  ifcif 4223  {csn 4314  cmpt2 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-v 3351  df-dif 3724  df-if 4224  df-sn 4315  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  smadiadetglem1  20695
  Copyright terms: Public domain W3C validator