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Theorem unixpm 5164
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 4735 . . 3 Rel (𝐴 × 𝐵)
2 relfld 5157 . . 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 266 . . . 4 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ (∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵))
5 xpm 5050 . . . 4 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
64, 5bitri 184 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7 dmxpm 4847 . . . 4 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8 rnxpm 5058 . . . 4 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9 uneq12 3284 . . . 4 ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
107, 8, 9syl2an 289 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
116, 10sylbir 135 . 2 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴𝐵))
123, 11eqtrid 2222 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  cun 3127   cuni 3809   × cxp 4624  dom cdm 4626  ran crn 4627  Rel wrel 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637
This theorem is referenced by: (None)
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