Theorem mposnif 6046

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}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem mposnif

Step	Hyp	Ref	Expression
1		elsni 3652	. . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋)
2	1	adantr 276	. . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋)
3	2	iftrued 3579	. 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶)
4	3	mpoeq3ia 6017	1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ifcif 3572  {csn 3634   ∈ cmpo 5953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-if 3573  df-sn 3640  df-oprab 5955  df-mpo 5956

This theorem is referenced by: (None)