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Theorem ssxpbm 4978
 Description: A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
ssxpbm (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem ssxpbm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpm 4964 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2 dmxpm 4763 . . . . . . . . 9 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
32adantl 275 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) = 𝐴)
41, 3sylbir 134 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
54adantr 274 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6 dmss 4742 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
76adantl 275 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
85, 7eqsstrrd 3135 . . . . 5 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9 dmxpss 4973 . . . . 5 dom (𝐶 × 𝐷) ⊆ 𝐶
108, 9sstrdi 3110 . . . 4 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴𝐶)
11 rnxpm 4972 . . . . . . . . 9 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
1211adantr 274 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) → ran (𝐴 × 𝐵) = 𝐵)
131, 12sylbir 134 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ran (𝐴 × 𝐵) = 𝐵)
1413adantr 274 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15 rnss 4773 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1615adantl 275 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1714, 16eqsstrrd 3135 . . . . 5 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18 rnxpss 4974 . . . . 5 ran (𝐶 × 𝐷) ⊆ 𝐷
1917, 18sstrdi 3110 . . . 4 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵𝐷)
2010, 19jca 304 . . 3 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴𝐶𝐵𝐷))
2120ex 114 . 2 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴𝐶𝐵𝐷)))
22 xpss12 4650 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
2321, 22impbid1 141 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ⊆ wss 3072   × cxp 4541  dom cdm 4543  ran crn 4544 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-rel 4550  df-cnv 4551  df-dm 4553  df-rn 4554 This theorem is referenced by:  xp11m  4981
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