MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssxpb Structured version   Visualization version   GIF version

Theorem ssxpb 6194
Description: A Cartesian product subclass relationship is equivalent to the conjunction of the analogous relationships for the factors. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
ssxpb ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))

Proof of Theorem ssxpb
StepHypRef Expression
1 xpnz 6179 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2 dmxp 5939 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
32adantl 481 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
41, 3sylbir 235 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
54adantr 480 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6 dmss 5913 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
76adantl 481 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
85, 7eqsstrrd 4019 . . . . 5 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9 dmxpss 6191 . . . . 5 dom (𝐶 × 𝐷) ⊆ 𝐶
108, 9sstrdi 3996 . . . 4 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴𝐶)
11 rnxp 6190 . . . . . . . . 9 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
1211adantr 480 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
131, 12sylbir 235 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
1413adantr 480 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15 rnss 5950 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1615adantl 481 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1714, 16eqsstrrd 4019 . . . . 5 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18 rnxpss 6192 . . . . 5 ran (𝐶 × 𝐷) ⊆ 𝐷
1917, 18sstrdi 3996 . . . 4 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵𝐷)
2010, 19jca 511 . . 3 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴𝐶𝐵𝐷))
2120ex 412 . 2 ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴𝐶𝐵𝐷)))
22 xpss12 5700 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
2321, 22impbid1 225 1 ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wne 2940  wss 3951  c0 4333   × cxp 5683  dom cdm 5685  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  xp11  6195  dibord  41161  aks6d1c2lem4  42128  aks6d1c2  42131
  Copyright terms: Public domain W3C validator