Theorem mpodifsnif 7475

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 4727	. . . 4 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋)
2	1	adantr 482	. . 3 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → ¬ 𝑖 = 𝑋)
3	2	iffalsed 4468	. 2 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷)
4	3	mpoeq3ia 7438	1 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ∖ cdif 3882  ifcif 4457  {csn 4558   ∈ cmpo 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-dif 3888  df-if 4458  df-sn 4559  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  smadiadetglem1  22658