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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 4643 | . . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋) | 
| 3 | 2 | iftrued 4533 | . 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 1540 ∈ wcel 2108 ifcif 4525 {csn 4626 ∈ cmpo 7433 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 df-sn 4627 df-oprab 7435 df-mpo 7436 | 
| This theorem is referenced by: mdetrsca2 22610 mdetrlin2 22613 mdetunilem5 22622 smadiadetglem2 22678 | 
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