Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2snif Structured version   Visualization version   GIF version

Theorem mpt2snif 6652
 Description: A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
Assertion
Ref Expression
mpt2snif (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)

Proof of Theorem mpt2snif
StepHypRef Expression
1 elsni 4142 . . . 4 (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
21adantr 480 . . 3 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → 𝑖 = 𝑋)
32iftrued 4044 . 2 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
43mpt2eq3ia 6618 1 (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  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-v 3175  df-if 4037  df-sn 4126  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  mdetrsca2  20229  mdetrlin2  20232  mdetunilem5  20241  smadiadetglem2  20297
 Copyright terms: Public domain W3C validator