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Theorem unixpm 5114
 Description: The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
unixpm (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unixpm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4688 . . 3 Rel (𝐴 × 𝐵)
2 relfld 5107 . . 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 ancom 264 . . . 4 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ (∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵))
5 xpm 5000 . . . 4 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
64, 5bitri 183 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7 dmxpm 4799 . . . 4 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8 rnxpm 5008 . . . 4 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9 uneq12 3252 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
107, 8, 9syl2an 287 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
116, 10sylbir 134 . 2 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
123, 11syl5eq 2199 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2125   ∪ cun 3096  ∪ cuni 3768   × cxp 4577  dom cdm 4579  ran crn 4580  Rel wrel 4584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-xp 4585  df-rel 4586  df-cnv 4587  df-dm 4589  df-rn 4590 This theorem is referenced by: (None)
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