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Theorem mpt2snif 6920
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 4338 . . . 4 (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
21adantr 472 . . 3 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → 𝑖 = 𝑋)
32iftrued 4238 . 2 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
43mpt2eq3ia 6886 1 (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  ifcif 4230  {csn 4321  cmpt2 6816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-if 4231  df-sn 4322  df-oprab 6818  df-mpt2 6819
This theorem is referenced by:  mdetrsca2  20632  mdetrlin2  20635  mdetunilem5  20644  smadiadetglem2  20700
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