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Theorem unixpm 5201
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 4768 . . 3 Rel (𝐴 × 𝐵)
2 relfld 5194 . . 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 5087 . . . 4 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
64, 5bitri 184 . . 3 ((∃𝑏 𝑏𝐵 ∧ ∃𝑎 𝑎𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7 dmxpm 4882 . . . 4 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8 rnxpm 5095 . . . 4 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9 uneq12 3308 . . . 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 2238 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wex 1503  wcel 2164  cun 3151   cuni 3835   × cxp 4657  dom cdm 4659  ran crn 4660  Rel wrel 4664
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670
This theorem is referenced by: (None)
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