ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpima2m GIF version

Theorem xpima2m 4846
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2m (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem xpima2m
StepHypRef Expression
1 df-ima 4426 . . . 4 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2 df-res 4425 . . . . 5 ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
32rneqi 4633 . . . 4 ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4 inxp 4540 . . . . 5 ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴𝐶) × (𝐵 ∩ V))
54rneqi 4633 . . . 4 ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴𝐶) × (𝐵 ∩ V))
61, 3, 53eqtri 2109 . . 3 ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴𝐶) × (𝐵 ∩ V))
7 rnxpm 4828 . . 3 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ran ((𝐴𝐶) × (𝐵 ∩ V)) = (𝐵 ∩ V))
86, 7syl5eq 2129 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = (𝐵 ∩ V))
9 inv1 3307 . 2 (𝐵 ∩ V) = 𝐵
108, 9syl6eq 2133 1 (∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wex 1424  wcel 1436  Vcvv 2615  cin 2987   × cxp 4411  ran crn 4414  cres 4415  cima 4416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by:  xpimasn  4847
  Copyright terms: Public domain W3C validator