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Theorem ssxp2 4701
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp2 (x x 𝐶 → ((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 4695 . . . . . 6 (x x 𝐶 → ran (𝐶 × A) = A)
21adantr 261 . . . . 5 ((x x 𝐶 (𝐶 × A) ⊆ (𝐶 × B)) → ran (𝐶 × A) = A)
3 rnss 4507 . . . . . 6 ((𝐶 × A) ⊆ (𝐶 × B) → ran (𝐶 × A) ⊆ ran (𝐶 × B))
43adantl 262 . . . . 5 ((x x 𝐶 (𝐶 × A) ⊆ (𝐶 × B)) → ran (𝐶 × A) ⊆ ran (𝐶 × B))
52, 4eqsstr3d 2974 . . . 4 ((x x 𝐶 (𝐶 × A) ⊆ (𝐶 × B)) → A ⊆ ran (𝐶 × B))
6 rnxpss 4697 . . . 4 ran (𝐶 × B) ⊆ B
75, 6syl6ss 2951 . . 3 ((x x 𝐶 (𝐶 × A) ⊆ (𝐶 × B)) → AB)
87ex 108 . 2 (x x 𝐶 → ((𝐶 × A) ⊆ (𝐶 × B) → AB))
9 xpss2 4392 . 2 (AB → (𝐶 × A) ⊆ (𝐶 × B))
108, 9impbid1 130 1 (x x 𝐶 → ((𝐶 × A) ⊆ (𝐶 × B) ↔ AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wss 2911   × cxp 4286  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  xpcanm  4703
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