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Theorem ssxpbm 5082
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 5068 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2 dmxpm 4865 . . . . . . . . 9 (∃𝑏 𝑏𝐵 → dom (𝐴 × 𝐵) = 𝐴)
32adantl 277 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) → dom (𝐴 × 𝐵) = 𝐴)
41, 3sylbir 135 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
54adantr 276 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6 dmss 4844 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
76adantl 277 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
85, 7eqsstrrd 3207 . . . . 5 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9 dmxpss 5077 . . . . 5 dom (𝐶 × 𝐷) ⊆ 𝐶
108, 9sstrdi 3182 . . . 4 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴𝐶)
11 rnxpm 5076 . . . . . . . . 9 (∃𝑎 𝑎𝐴 → ran (𝐴 × 𝐵) = 𝐵)
1211adantr 276 . . . . . . . 8 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) → ran (𝐴 × 𝐵) = 𝐵)
131, 12sylbir 135 . . . . . . 7 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ran (𝐴 × 𝐵) = 𝐵)
1413adantr 276 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15 rnss 4875 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1615adantl 277 . . . . . 6 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
1714, 16eqsstrrd 3207 . . . . 5 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18 rnxpss 5078 . . . . 5 ran (𝐶 × 𝐷) ⊆ 𝐷
1917, 18sstrdi 3182 . . . 4 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵𝐷)
2010, 19jca 306 . . 3 ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴𝐶𝐵𝐷))
2120ex 115 . 2 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴𝐶𝐵𝐷)))
22 xpss12 4751 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
2321, 22impbid1 142 1 (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wss 3144   × cxp 4642  dom cdm 4644  ran crn 4645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655
This theorem is referenced by:  xp11m  5085
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