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Theorem unixp 5829
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 5283 . . 3 Rel (𝐴 × 𝐵)
2 relfld 5822 . . 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 5286 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 5710 . . . . 5 (𝐴 × ∅) = ∅
64, 5syl6eq 2810 . . . 4 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
76necon3i 2964 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8 xpeq1 5280 . . . . 5 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9 0xp 5356 . . . . 5 (∅ × 𝐵) = ∅
108, 9syl6eq 2810 . . . 4 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
1110necon3i 2964 . . 3 ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12 dmxp 5499 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13 rnxp 5722 . . . 4 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14 uneq12 3905 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
1512, 13, 14syl2an 495 . . 3 ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
167, 11, 15syl2anc 696 . 2 ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
173, 16syl5eq 2806 1 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wne 2932  cun 3713  c0 4058   cuni 4588   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  unixpid  5831  rankxpl  8913  rankxplim2  8918  rankxplim3  8919  rankxpsuc  8920
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