Theorem mpodifsnif 7567

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 4816	. . . 4 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
2	1	adantr 480	. . 3 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → ¬ 𝑖 = 𝑋)
3	2	iffalsed 4559	. 2 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
4	3	mpoeq3ia 7530	1 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  ifcif 4548  {csn 4648   ∈ cmpo 7452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-if 4549  df-sn 4649  df-oprab 7454  df-mpo 7455

This theorem is referenced by:  smadiadetglem1  22700