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Theorem ssxpb 6173
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 6158 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2 dmxp 5928 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
32adantl 482 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
41, 3sylbir 234 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
54adantr 481 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6 dmss 5902 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
76adantl 482 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
85, 7eqsstrrd 4021 . . . . 5 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9 dmxpss 6170 . . . . 5 dom (𝐶 × 𝐷) ⊆ 𝐶
108, 9sstrdi 3994 . . . 4 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴𝐶)
11 rnxp 6169 . . . . . . . . 9 (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
1211adantr 481 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
131, 12sylbir 234 . . . . . . 7 ((𝐴 × 𝐵) ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
1413adantr 481 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15 rnss 5938 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1615adantl 482 . . . . . 6 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1714, 16eqsstrrd 4021 . . . . 5 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18 rnxpss 6171 . . . . 5 ran (𝐶 × 𝐷) ⊆ 𝐷
1917, 18sstrdi 3994 . . . 4 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵𝐷)
2010, 19jca 512 . . 3 (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴𝐶𝐵𝐷))
2120ex 413 . 2 ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴𝐶𝐵𝐷)))
22 xpss12 5691 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
2321, 22impbid1 224 1 ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wne 2940  wss 3948  c0 4322   × cxp 5674  dom cdm 5676  ran crn 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687
This theorem is referenced by:  xp11  6174  dibord  40025
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