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Theorem unixpm 4796
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 4390 . . 3 Rel (A × B)
2 relfld 4789 . . 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 253 . . . 4 ((𝑏 𝑏 B 𝑎 𝑎 A) ↔ (𝑎 𝑎 A 𝑏 𝑏 B))
5 xpm 4688 . . . 4 ((𝑎 𝑎 A 𝑏 𝑏 B) ↔ x x (A × B))
64, 5bitri 173 . . 3 ((𝑏 𝑏 B 𝑎 𝑎 A) ↔ x x (A × B))
7 dmxpm 4498 . . . 4 (𝑏 𝑏 B → dom (A × B) = A)
8 rnxpm 4695 . . . 4 (𝑎 𝑎 A → ran (A × B) = B)
9 uneq12 3086 . . . 4 ((dom (A × B) = A ran (A × B) = B) → (dom (A × B) ∪ ran (A × B)) = (AB))
107, 8, 9syl2an 273 . . 3 ((𝑏 𝑏 B 𝑎 𝑎 A) → (dom (A × B) ∪ ran (A × B)) = (AB))
116, 10sylbir 125 . 2 (x x (A × B) → (dom (A × B) ∪ ran (A × B)) = (AB))
123, 11syl5eq 2081 1 (x x (A × B) → (A × B) = (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  cun 2909   cuni 3571   × cxp 4286  dom cdm 4288  ran crn 4289  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by: (None)
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