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Mirrors > Home > ILE Home > Th. List > xpima1 | GIF version |
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Ref | Expression |
---|---|
xpima1 | ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4358 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶) | |
2 | df-res 4357 | . . . 4 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
3 | 2 | rneqi 4562 | . . 3 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) |
4 | inxp 4470 | . . . 4 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
5 | 4 | rneqi 4562 | . . 3 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) |
6 | 1, 3, 5 | 3eqtri 2064 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) |
7 | xpeq1 4359 | . . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V))) | |
8 | 0xp 4420 | . . . 4 ⊢ (∅ × (𝐵 ∩ V)) = ∅ | |
9 | 7, 8 | syl6eq 2088 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅) |
10 | rneq 4561 | . . . 4 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ∅) | |
11 | rn0 4588 | . . . 4 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | syl6eq 2088 | . . 3 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅) |
13 | 9, 12 | syl 14 | . 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅) |
14 | 6, 13 | syl5eq 2084 | 1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 Vcvv 2557 ∩ cin 2916 ∅c0 3224 × cxp 4343 ran crn 4346 ↾ cres 4347 “ cima 4348 |
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-in1 544 ax-in2 545 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-fal 1249 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-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 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 df-res 4357 df-ima 4358 |
This theorem is referenced by: (None) |
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