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Theorem unixp 6313
Description: The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unixp ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))

Proof of Theorem unixp
StepHypRef Expression
1 relxp 5718 . . 3 Rel (𝐴 × 𝐵)
2 relfld 6306 . . 3 (Rel (𝐴 × 𝐵) → (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))
31, 2ax-mp 5 . 2 (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))
4 xpeq2 5721 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 6189 . . . . 5 (𝐴 × ∅) = ∅
64, 5eqtrdi 2796 . . . 4 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
76necon3i 2979 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8 xpeq1 5714 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9 0xp 5798 . . . . 5 (∅ × 𝐵) = ∅
108, 9eqtrdi 2796 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
1110necon3i 2979 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12 dmxp 5953 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13 rnxp 6201 . . . 4 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14 uneq12 4186 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
1512, 13, 14syl2an 595 . . 3 ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
167, 11, 15syl2anc 583 . 2 ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
173, 16eqtrid 2792 1 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wne 2946  cun 3974  c0 4352   cuni 4931   × cxp 5698  dom cdm 5700  ran crn 5701  Rel wrel 5705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711
This theorem is referenced by:  unixpid  6315  rankxpl  9944  rankxplim2  9949  rankxplim3  9950  rankxpsuc  9951
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