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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 4078 | . . 3 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ V) | |
3 | xpexg 4623 | . . 3 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V) | |
4 | 2, 3 | sylan2 284 | . 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐼 × {𝑋}) ∈ V) |
5 | sneq 3508 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
6 | 5 | xpeq2d 4533 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋})) |
7 | fdiagfn.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
8 | 6, 7 | fvmptg 5465 | . 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 1316 ∈ wcel 1465 Vcvv 2660 {csn 3497 ↦ cmpt 3959 × cxp 4507 ‘cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: (None) |
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