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Theorem mposnif 5831
 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
mposnif (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)

Proof of Theorem mposnif
StepHypRef Expression
1 elsni 3513 . . . 4 (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
21adantr 272 . . 3 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → 𝑖 = 𝑋)
32iftrued 3449 . 2 ((𝑖 ∈ {𝑋} ∧ 𝑗𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
43mpoeq3ia 5802 1 (𝑖 ∈ {𝑋}, 𝑗𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1314   ∈ wcel 1463  ifcif 3442  {csn 3495   ∈ cmpo 5742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-if 3443  df-sn 3501  df-oprab 5744  df-mpo 5745 This theorem is referenced by: (None)
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