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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 5572 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
2 | relfld 6125 | . . 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 5575 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
5 | xp0 6014 | . . . . 5 ⊢ (𝐴 × ∅) = ∅ | |
6 | 4, 5 | syl6eq 2872 | . . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
7 | 6 | necon3i 3048 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅) |
8 | xpeq1 5568 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
9 | 0xp 5648 | . . . . 5 ⊢ (∅ × 𝐵) = ∅ | |
10 | 8, 9 | syl6eq 2872 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) |
11 | 10 | necon3i 3048 | . . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅) |
12 | dmxp 5798 | . . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
13 | rnxp 6026 | . . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) | |
14 | uneq12 4133 | . . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) | |
15 | 12, 13, 14 | syl2an 597 | . . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
16 | 7, 11, 15 | syl2anc 586 | . 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵)) |
17 | 3, 16 | syl5eq 2868 | 1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 3016 ∪ cun 3933 ∅c0 4290 ∪ cuni 4837 × cxp 5552 dom cdm 5554 ran crn 5555 Rel wrel 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 |
This theorem is referenced by: unixpid 6134 rankxpl 9303 rankxplim2 9308 rankxplim3 9309 rankxpsuc 9310 |
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