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Mirrors > Home > ILE Home > Th. List > rnxpid | GIF version |
Description: The range of a square cross product. (Contributed by FL, 17-May-2010.) |
Ref | Expression |
---|---|
rnxpid | ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnxpss 4754 | . 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴 | |
2 | opelxp 4374 | . . . . . 6 ⊢ (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
3 | anidm 376 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
4 | 2, 3 | bitri 173 | . . . . 5 ⊢ (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴) |
5 | opeq1 3549 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 〈𝑥, 𝑥〉 = 〈𝑦, 𝑥〉) | |
6 | 5 | eleq1d 2106 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴))) |
7 | 6 | equcoms 1594 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) ↔ 〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴))) |
8 | 7 | biimpd 132 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) → 〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴))) |
9 | 8 | spimev 1741 | . . . . 5 ⊢ (〈𝑥, 𝑥〉 ∈ (𝐴 × 𝐴) → ∃𝑦〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴)) |
10 | 4, 9 | sylbir 125 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴)) |
11 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | elrn2 4576 | . . . 4 ⊢ (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦〈𝑦, 𝑥〉 ∈ (𝐴 × 𝐴)) |
13 | 10, 12 | sylibr 137 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (𝐴 × 𝐴)) |
14 | 13 | ssriv 2949 | . 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴) |
15 | 1, 14 | eqssi 2961 | 1 ⊢ ran (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 〈cop 3378 × cxp 4343 ran crn 4346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: (None) |
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