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Mirrors > Home > ILE Home > Th. List > fvdiagfn | GIF version |
Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
fdiagfn.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) |
Ref | Expression |
---|---|
fvdiagfn | ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
2 | snexg 4108 | . . 3 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ V) | |
3 | xpexg 4653 | . . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V) | |
4 | 2, 3 | sylan2 284 | . 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐼 × {𝑋}) ∈ V) |
5 | sneq 3538 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
6 | 5 | xpeq2d 4563 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋})) |
7 | fdiagfn.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
8 | 6, 7 | fvmptg 5497 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝐼 × {𝑋}) ∈ V) → (𝐹‘𝑋) = (𝐼 × {𝑋})) |
9 | 1, 4, 8 | syl2anc 408 | 1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 ↦ cmpt 3989 × cxp 4537 ‘cfv 5123 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: (None) |
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