Theorem mpodifsnif 5864

Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)

Assertion

Ref	Expression
mpodifsnif	⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)

Proof of Theorem mpodifsnif

Step	Hyp	Ref	Expression
1		eldifsn 3650	. . . . 5 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑖 ∈ 𝐴 ∧ 𝑖 ≠ 𝑋))
2		neneq 2330	. . . . 5 ⊢ (𝑖 ≠ 𝑋 → ¬ 𝑖 = 𝑋)
3	1, 2	simplbiim 384	. . . 4 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
4	3	adantr 274	. . 3 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → ¬ 𝑖 = 𝑋)
5	4	iffalsed 3484	. 2 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
6	5	mpoeq3ia 5836	1 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ≠ wne 2308   ∖ cdif 3068  ifcif 3474  {csn 3527   ∈ cmpo 5776

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-if 3475  df-sn 3533  df-oprab 5778  df-mpo 5779

This theorem is referenced by: (None)