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Theorem ssxp2 4836
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp2 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssxp2
StepHypRef Expression
1 rnxpm 4828 . . . . . 6 (∃𝑥 𝑥𝐶 → ran (𝐶 × 𝐴) = 𝐴)
21adantr 270 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → ran (𝐶 × 𝐴) = 𝐴)
3 rnss 4635 . . . . . 6 ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) → ran (𝐶 × 𝐴) ⊆ ran (𝐶 × 𝐵))
43adantl 271 . . . . 5 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → ran (𝐶 × 𝐴) ⊆ ran (𝐶 × 𝐵))
52, 4eqsstr3d 3050 . . . 4 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → 𝐴 ⊆ ran (𝐶 × 𝐵))
6 rnxpss 4830 . . . 4 ran (𝐶 × 𝐵) ⊆ 𝐵
75, 6syl6ss 3026 . . 3 ((∃𝑥 𝑥𝐶 ∧ (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) → 𝐴𝐵)
87ex 113 . 2 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) → 𝐴𝐵))
9 xpss2 4519 . 2 (𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
108, 9impbid1 140 1 (∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  wss 2988   × cxp 4411  ran crn 4414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424
This theorem is referenced by:  xpcanm  4838
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