ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssxp2 GIF version

Theorem ssxp2 4788
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp2 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssxp2
StepHypRef Expression
1 rnxpm 4782 . . . . . 6 (∃𝑥 𝑥𝐶 → ran (𝐶 × 𝐴) = 𝐴)
21adantr 270 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → ran (𝐶 × 𝐴) = 𝐴)
3 rnss 4592 . . . . . 6 ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) → ran (𝐶 × 𝐴) ⊆ ran (𝐶 × 𝐵))
43adantl 271 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → ran (𝐶 × 𝐴) ⊆ ran (𝐶 × 𝐵))
52, 4eqsstr3d 3035 . . . 4 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → 𝐴 ⊆ ran (𝐶 × 𝐵))
6 rnxpss 4784 . . . 4 ran (𝐶 × 𝐵) ⊆ 𝐵
75, 6syl6ss 3012 . . 3 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → 𝐴𝐵)
87ex 113 . 2 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) → 𝐴𝐵))
9 xpss2 4477 . 2 (𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
108, 9impbid1 140 1 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wss 2974   × cxp 4369  ran crn 4372
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  xpcanm  4790
  Copyright terms: Public domain W3C validator