ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpima1 GIF version

Theorem xpima1 4955
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 4522 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4521 . . . 4 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4737 . . 3 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4643 . . . 4 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4737 . . 3 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2142 . 2 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 xpeq1 4523 . . . 4 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8 0xp 4589 . . . 4 (∅ × (𝐵 ∩ V)) = ∅
97, 8syl6eq 2166 . . 3 ((𝐴𝐶) = ∅ → ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
10 rneq 4736 . . . 4 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ran ∅)
11 rn0 4765 . . . 4 ran ∅ = ∅
1210, 11syl6eq 2166 . . 3 (((𝐴𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
139, 12syl 14 . 2 ((𝐴𝐶) = ∅ → ran ((𝐴𝐶) × (𝐵 ∩ V)) = ∅)
146, 13syl5eq 2162 1 ((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  Vcvv 2660  cin 3040  c0 3333   × cxp 4507  ran crn 4510  cres 4511  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator