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Mirrors > Home > ILE Home > Th. List > mposnif | 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 3550 | . . . 4 ⊢ (𝑖 ∈ {𝑋} → 𝑖 = 𝑋) | |
2 | 1 | adantr 274 | . . 3 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → 𝑖 = 𝑋) |
3 | 2 | iftrued 3486 | . 2 ⊢ ((𝑖 ∈ {𝑋} ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐶) |
4 | 3 | mpoeq3ia 5844 | 1 ⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 ifcif 3479 {csn 3532 ∈ cmpo 5784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-if 3480 df-sn 3538 df-oprab 5786 df-mpo 5787 |
This theorem is referenced by: (None) |
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