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Theorem unixp 5886
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 5329 . . 3 Rel (𝐴 × 𝐵)
2 relfld 5879 . . 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 5332 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 5768 . . . . 5 (𝐴 × ∅) = ∅
64, 5syl6eq 2848 . . . 4 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
76necon3i 3002 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8 xpeq1 5325 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9 0xp 5403 . . . . 5 (∅ × 𝐵) = ∅
108, 9syl6eq 2848 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
1110necon3i 3002 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12 dmxp 5546 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13 rnxp 5780 . . . 4 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14 uneq12 3959 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
1512, 13, 14syl2an 590 . . 3 ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
167, 11, 15syl2anc 580 . 2 ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
173, 16syl5eq 2844 1 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2970  cun 3766  c0 4114   cuni 4627   × cxp 5309  dom cdm 5311  ran crn 5312  Rel wrel 5316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322
This theorem is referenced by:  unixpid  5888  rankxpl  8987  rankxplim2  8992  rankxplim3  8993  rankxpsuc  8994
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