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Theorem rninxp 4706
Description: Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rninxp (ran (𝐶 ∩ (A × B)) = By B x A x𝐶y)
Distinct variable groups:   x,y,A   y,B   x,𝐶,y
Allowed substitution hint:   B(x)

Proof of Theorem rninxp
StepHypRef Expression
1 dfss3 2929 . 2 (B ⊆ ran (𝐶A) ↔ y B y ran (𝐶A))
2 ssrnres 4705 . 2 (B ⊆ ran (𝐶A) ↔ ran (𝐶 ∩ (A × B)) = B)
3 df-ima 4300 . . . . 5 (𝐶A) = ran (𝐶A)
43eleq2i 2101 . . . 4 (y (𝐶A) ↔ y ran (𝐶A))
5 vex 2554 . . . . 5 y V
65elima 4615 . . . 4 (y (𝐶A) ↔ x A x𝐶y)
74, 6bitr3i 175 . . 3 (y ran (𝐶A) ↔ x A x𝐶y)
87ralbii 2324 . 2 (y B y ran (𝐶A) ↔ y B x A x𝐶y)
91, 2, 83bitr3i 199 1 (ran (𝐶 ∩ (A × B)) = By B x A x𝐶y)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301  cin 2910  wss 2911   class class class wbr 3754   × cxp 4285  ran crn 4288  cres 4289  cima 4290
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:  dminxp  4707  fncnv  4906
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