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Theorem dminxp 4707
 Description: Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
dminxp (dom (𝐶 ∩ (A × B)) = Ax A y B x𝐶y)
Distinct variable groups:   x,A   x,y,B   x,𝐶,y
Allowed substitution hint:   A(y)

Proof of Theorem dminxp
StepHypRef Expression
1 dfdm4 4469 . . . 4 dom (𝐶 ∩ (A × B)) = ran (𝐶 ∩ (A × B))
2 cnvin 4673 . . . . . 6 (𝐶 ∩ (A × B)) = (𝐶(A × B))
3 cnvxp 4684 . . . . . . 7 (A × B) = (B × A)
43ineq2i 3129 . . . . . 6 (𝐶(A × B)) = (𝐶 ∩ (B × A))
52, 4eqtri 2057 . . . . 5 (𝐶 ∩ (A × B)) = (𝐶 ∩ (B × A))
65rneqi 4504 . . . 4 ran (𝐶 ∩ (A × B)) = ran (𝐶 ∩ (B × A))
71, 6eqtri 2057 . . 3 dom (𝐶 ∩ (A × B)) = ran (𝐶 ∩ (B × A))
87eqeq1i 2044 . 2 (dom (𝐶 ∩ (A × B)) = A ↔ ran (𝐶 ∩ (B × A)) = A)
9 rninxp 4706 . 2 (ran (𝐶 ∩ (B × A)) = Ax A y B y𝐶x)
10 vex 2554 . . . . 5 y V
11 vex 2554 . . . . 5 x V
1210, 11brcnv 4460 . . . 4 (y𝐶xx𝐶y)
1312rexbii 2325 . . 3 (y B y𝐶xy B x𝐶y)
1413ralbii 2324 . 2 (x A y B y𝐶xx A y B x𝐶y)
158, 9, 143bitri 195 1 (dom (𝐶 ∩ (A × B)) = Ax A y B x𝐶y)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1242  ∀wral 2300  ∃wrex 2301   ∩ cin 2910   class class class wbr 3754   × cxp 4285  ◡ccnv 4286  dom cdm 4287  ran crn 4288 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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934 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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-rel 4294  df-cnv 4295  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300 This theorem is referenced by: (None)
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