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Theorem xpima1 4710
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima1 ((A𝐶) = ∅ → ((A × B) “ 𝐶) = ∅)

Proof of Theorem xpima1
StepHypRef Expression
1 df-ima 4301 . . 3 ((A × B) “ 𝐶) = ran ((A × B) ↾ 𝐶)
2 df-res 4300 . . . 4 ((A × B) ↾ 𝐶) = ((A × B) ∩ (𝐶 × V))
32rneqi 4505 . . 3 ran ((A × B) ↾ 𝐶) = ran ((A × B) ∩ (𝐶 × V))
4 inxp 4413 . . . 4 ((A × B) ∩ (𝐶 × V)) = ((A𝐶) × (B ∩ V))
54rneqi 4505 . . 3 ran ((A × B) ∩ (𝐶 × V)) = ran ((A𝐶) × (B ∩ V))
61, 3, 53eqtri 2061 . 2 ((A × B) “ 𝐶) = ran ((A𝐶) × (B ∩ V))
7 xpeq1 4302 . . . 4 ((A𝐶) = ∅ → ((A𝐶) × (B ∩ V)) = (∅ × (B ∩ V)))
8 0xp 4363 . . . 4 (∅ × (B ∩ V)) = ∅
97, 8syl6eq 2085 . . 3 ((A𝐶) = ∅ → ((A𝐶) × (B ∩ V)) = ∅)
10 rneq 4504 . . . 4 (((A𝐶) × (B ∩ V)) = ∅ → ran ((A𝐶) × (B ∩ V)) = ran ∅)
11 rn0 4531 . . . 4 ran ∅ = ∅
1210, 11syl6eq 2085 . . 3 (((A𝐶) × (B ∩ V)) = ∅ → ran ((A𝐶) × (B ∩ V)) = ∅)
139, 12syl 14 . 2 ((A𝐶) = ∅ → ran ((A𝐶) × (B ∩ V)) = ∅)
146, 13syl5eq 2081 1 ((A𝐶) = ∅ → ((A × B) “ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  Vcvv 2551  cin 2910  c0 3218   × cxp 4286  ran crn 4289  cres 4290  cima 4291
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-in1 544  ax-in2 545  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 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  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  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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