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Theorem unixpm 4934
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 4517 . . 3 Rel (𝐴 × 𝐵)
2 relfld 4927 . . 3 (Rel (𝐴 × 𝐵) → (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))
31, 2ax-mp 7 . 2 (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))
4 ancom 262 . . . 4 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ (∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵))
5 xpm 4821 . . . 4 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
64, 5bitri 182 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7 dmxpm 4626 . . . 4 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8 rnxpm 4828 . . . 4 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9 uneq12 3138 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
107, 8, 9syl2an 283 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
116, 10sylbir 133 . 2 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
123, 11syl5eq 2129 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wex 1424  wcel 1436  cun 2986   cuni 3638   × cxp 4411  dom cdm 4413  ran crn 4414  Rel wrel 4418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424
This theorem is referenced by: (None)
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