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Mirrors > Home > MPE Home > Th. List > unixp | Structured version Visualization version GIF version |
Description: The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
unixp | ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5696 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
2 | relfld 6279 | . . 3 ⊢ (Rel (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) |
4 | xpeq2 5699 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
5 | xp0 6162 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2784 | . . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
7 | 6 | necon3i 2970 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅) |
8 | xpeq1 5692 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
9 | 0xp 5776 | . . . . 5 ⊢ (∅ × 𝐵) = ∅ | |
10 | 8, 9 | eqtrdi 2784 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) |
11 | 10 | necon3i 2970 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅) |
12 | dmxp 5931 | . . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
13 | rnxp 6174 | . . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
14 | uneq12 4157 | . . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) | |
15 | 12, 13, 14 | syl2an 595 | . . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
16 | 7, 11, 15 | syl2anc 583 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
17 | 3, 16 | eqtrid 2780 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ≠ wne 2937 ∪ cun 3945 ∅c0 4323 ∪ cuni 4908 × cxp 5676 dom cdm 5678 ran crn 5679 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: unixpid 6288 rankxpl 9898 rankxplim2 9903 rankxplim3 9904 rankxpsuc 9905 |
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