Theorem mpodifsnif 7515

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		eldifsnneq 4786	. . . 4 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
2	1	adantr 480	. . 3 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → ¬ 𝑖 = 𝑋)
3	2	iffalsed 4531	. 2 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
4	3	mpoeq3ia 7479	1 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ∖ cdif 3937  ifcif 4520  {csn 4620   ∈ cmpo 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3943  df-if 4521  df-sn 4621  df-oprab 7405  df-mpo 7406

This theorem is referenced by:  smadiadetglem1  22495