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

Theorem mpt2snif 6739
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 4185 . . . 4 (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
21adantr 481 . . 3 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → 𝑖 = 𝑋)
32iftrued 4085 . 2 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
43mpt2eq3ia 6705 1 (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  wcel 1988  ifcif 4077  {csn 4168  cmpt2 6637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-if 4078  df-sn 4169  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  mdetrsca2  20391  mdetrlin2  20394  mdetunilem5  20403  smadiadetglem2  20459
  Copyright terms: Public domain W3C validator