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Theorem rnxpid 4805
Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid ran (𝐴 × 𝐴) = 𝐴

Proof of Theorem rnxpid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnxpss 4804 . 2 ran (𝐴 × 𝐴) ⊆ 𝐴
2 opelxp 4420 . . . . . 6 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑥𝐴))
3 anidm 388 . . . . . 6 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
42, 3bitri 182 . . . . 5 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ 𝑥𝐴)
5 opeq1 3590 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
65eleq1d 2151 . . . . . . . 8 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
76equcoms 1636 . . . . . . 7 (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
87biimpd 142 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
98spimev 1784 . . . . 5 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
104, 9sylbir 133 . . . 4 (𝑥𝐴 → ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
11 vex 2613 . . . . 5 𝑥 ∈ V
1211elrn2 4624 . . . 4 (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
1310, 12sylibr 132 . . 3 (𝑥𝐴𝑥 ∈ ran (𝐴 × 𝐴))
1413ssriv 3012 . 2 𝐴 ⊆ ran (𝐴 × 𝐴)
151, 14eqssi 3024 1 ran (𝐴 × 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  cop 3419   × cxp 4389  ran crn 4392
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by: (None)
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