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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 5607 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
2 | relfld 6178 | . . 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 5610 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
5 | xp0 6061 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2794 | . . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
7 | 6 | necon3i 2976 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅) |
8 | xpeq1 5603 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
9 | 0xp 5685 | . . . . 5 ⊢ (∅ × 𝐵) = ∅ | |
10 | 8, 9 | eqtrdi 2794 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) |
11 | 10 | necon3i 2976 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅) |
12 | dmxp 5838 | . . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
13 | rnxp 6073 | . . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
14 | uneq12 4092 | . . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) | |
15 | 12, 13, 14 | syl2an 596 | . . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
16 | 7, 11, 15 | syl2anc 584 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
17 | 3, 16 | eqtrid 2790 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ≠ wne 2943 ∪ cun 3885 ∅c0 4256 ∪ cuni 4839 × cxp 5587 dom cdm 5589 ran crn 5590 Rel wrel 5594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 |
This theorem is referenced by: unixpid 6187 rankxpl 9633 rankxplim2 9638 rankxplim3 9639 rankxpsuc 9640 |
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