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Theorem rnxpid 4698
 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 4697 . 2 ran (A × A) ⊆ A
2 opelxp 4317 . . . . . 6 (⟨x, x (A × A) ↔ (x A x A))
3 anidm 376 . . . . . 6 ((x A x A) ↔ x A)
42, 3bitri 173 . . . . 5 (⟨x, x (A × A) ↔ x A)
5 opeq1 3540 . . . . . . . . 9 (x = y → ⟨x, x⟩ = ⟨y, x⟩)
65eleq1d 2103 . . . . . . . 8 (x = y → (⟨x, x (A × A) ↔ ⟨y, x (A × A)))
76equcoms 1591 . . . . . . 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 1738 . . . . 5 (⟨x, x (A × A) → yy, x (A × A))
104, 9sylbir 125 . . . 4 (x Ayy, x (A × A))
11 vex 2554 . . . . 5 x V
1211elrn2 4519 . . . 4 (x ran (A × A) ↔ yy, x (A × A))
1310, 12sylibr 137 . . 3 (x Ax ran (A × A))
1413ssriv 2943 . 2 A ⊆ ran (A × A)
151, 14eqssi 2955 1 ran (A × A) = A
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370   × cxp 4286  ran crn 4289 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-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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