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Mirrors > Home > MPE Home > Th. List > mposnif | Structured version Visualization version GIF version |
Description: A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.) |
Ref | Expression |
---|---|
mposnif | ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4648 | . . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋) |
3 | 2 | iftrued 4539 | . 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶) |
4 | 3 | mpoeq3ia 7511 | 1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 {csn 4631 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 df-sn 4632 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: mdetrsca2 22626 mdetrlin2 22629 mdetunilem5 22638 smadiadetglem2 22694 |
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