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Theorem rnxpid 4755
 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 4754 . 2 ran (𝐴 × 𝐴) ⊆ 𝐴
2 opelxp 4374 . . . . . 6 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑥𝐴))
3 anidm 376 . . . . . 6 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
42, 3bitri 173 . . . . 5 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ 𝑥𝐴)
5 opeq1 3549 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
65eleq1d 2106 . . . . . . . 8 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
76equcoms 1594 . . . . . . 7 (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
87biimpd 132 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
98spimev 1741 . . . . 5 (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
104, 9sylbir 125 . . . 4 (𝑥𝐴 → ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
11 vex 2560 . . . . 5 𝑥 ∈ V
1211elrn2 4576 . . . 4 (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
1310, 12sylibr 137 . . 3 (𝑥𝐴𝑥 ∈ ran (𝐴 × 𝐴))
1413ssriv 2949 . 2 𝐴 ⊆ ran (𝐴 × 𝐴)
151, 14eqssi 2961 1 ran (𝐴 × 𝐴) = 𝐴
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   × cxp 4343  ran crn 4346 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356 This theorem is referenced by: (None)
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