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Theorem unixp 6312
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 5717 . . 3 Rel (𝐴 × 𝐵)
2 relfld 6305 . . 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 5720 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 6188 . . . . 5 (𝐴 × ∅) = ∅
64, 5eqtrdi 2790 . . . 4 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
76necon3i 2975 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8 xpeq1 5713 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9 0xp 5797 . . . . 5 (∅ × 𝐵) = ∅
108, 9eqtrdi 2790 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
1110necon3i 2975 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12 dmxp 5952 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13 rnxp 6200 . . . 4 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14 uneq12 4180 . . . 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 2786 1 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wne 2942  cun 3968  c0 4347   cuni 4931   × cxp 5697  dom cdm 5699  ran crn 5700  Rel wrel 5704
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 2105  ax-9 2113  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710
This theorem is referenced by:  unixpid  6314  rankxpl  9940  rankxplim2  9945  rankxplim3  9946  rankxpsuc  9947
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