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Mirrors > Home > MPE Home > Th. List > fvdiagfn | Structured version Visualization version 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 | fdiagfn.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
2 | sneq 4633 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | xpeq2d 5699 | . 2 ⊢ (𝑥 = 𝑋 → (𝐼 × {𝑥}) = (𝐼 × {𝑋})) |
4 | simpr 484 | . 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | snex 5424 | . . . 4 ⊢ {𝑋} ∈ V | |
6 | xpexg 7733 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐼 × {𝑋}) ∈ V) | |
7 | 5, 6 | mpan2 688 | . . 3 ⊢ (𝐼 ∈ 𝑊 → (𝐼 × {𝑋}) ∈ V) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐼 × {𝑋}) ∈ V) |
9 | 1, 3, 4, 8 | fvmptd3 7014 | 1 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ↦ cmpt 5224 × cxp 5667 ‘cfv 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 |
This theorem is referenced by: pwsdiagmhm 18753 pwsdiaglmhm 20902 |
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