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Theorem rnxpm 4695
 Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
rnxpm (x x A → ran (A × B) = B)
Distinct variable group:   x,A
Allowed substitution hint:   B(x)

Proof of Theorem rnxpm
StepHypRef Expression
1 df-rn 4299 . . 3 ran (A × B) = dom (A × B)
2 cnvxp 4685 . . . 4 (A × B) = (B × A)
32dmeqi 4479 . . 3 dom (A × B) = dom (B × A)
41, 3eqtri 2057 . 2 ran (A × B) = dom (B × A)
5 dmxpm 4498 . 2 (x x A → dom (B × A) = B)
64, 5syl5eq 2081 1 (x x A → ran (A × B) = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390   × cxp 4286  ◡ccnv 4287  dom cdm 4288  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:  ssxpbm  4699  ssxp2  4701  xpexr2m  4705  xpima2m  4711  unixpm  4796
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