![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4657 | . . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶) | |
2 | df-res 4656 | . . . 4 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
3 | 2 | rneqi 4873 | . . 3 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) |
4 | inxp 4779 | . . . 4 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
5 | 4 | rneqi 4873 | . . 3 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) |
6 | 1, 3, 5 | 3eqtri 2214 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) |
7 | xpeq1 4658 | . . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V))) | |
8 | 0xp 4724 | . . . 4 ⊢ (∅ × (𝐵 ∩ V)) = ∅ | |
9 | 7, 8 | eqtrdi 2238 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅) |
10 | rneq 4872 | . . . 4 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ∅) | |
11 | rn0 4901 | . . . 4 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | eqtrdi 2238 | . . 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 | eqtrid 2234 | 1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 Vcvv 2752 ∩ cin 3143 ∅c0 3437 × cxp 4642 ran crn 4645 ↾ cres 4646 “ cima 4647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |