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Theorem ssxp1 4978
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp1 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssxp1
StepHypRef Expression
1 dmxpm 4762 . . . . . 6 (∃𝑥 𝑥𝐶 → dom (𝐴 × 𝐶) = 𝐴)
21adantr 274 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) → dom (𝐴 × 𝐶) = 𝐴)
3 dmss 4741 . . . . . 6 ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) → dom (𝐴 × 𝐶) ⊆ dom (𝐵 × 𝐶))
43adantl 275 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) → dom (𝐴 × 𝐶) ⊆ dom (𝐵 × 𝐶))
52, 4eqsstrrd 3134 . . . 4 ((∃𝑥 𝑥𝐶 ∧ (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) → 𝐴 ⊆ dom (𝐵 × 𝐶))
6 dmxpss 4972 . . . 4 dom (𝐵 × 𝐶) ⊆ 𝐵
75, 6sstrdi 3109 . . 3 ((∃𝑥 𝑥𝐶 ∧ (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶)) → 𝐴𝐵)
87ex 114 . 2 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) → 𝐴𝐵))
9 xpss1 4652 . 2 (𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
108, 9impbid1 141 1 (∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wss 3071   × cxp 4540  dom cdm 4542
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-xp 4548  df-dm 4552
This theorem is referenced by:  xpcan2m  4982
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