Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cossxp2 GIF version

Theorem cossxp2 5062
 Description: The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
cossxp2.r (𝜑𝑅 ⊆ (𝐴 × 𝐵))
cossxp2.s (𝜑𝑆 ⊆ (𝐵 × 𝐶))
Assertion
Ref Expression
cossxp2 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))

Proof of Theorem cossxp2
StepHypRef Expression
1 cossxp 5061 . 2 (𝑆𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2 cossxp2.r . . . 4 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
3 dmxpss2 4971 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
42, 3syl 14 . . 3 (𝜑 → dom 𝑅𝐴)
5 cossxp2.s . . . 4 (𝜑𝑆 ⊆ (𝐵 × 𝐶))
6 rnxpss2 4972 . . . 4 (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆𝐶)
75, 6syl 14 . . 3 (𝜑 → ran 𝑆𝐶)
8 xpss12 4646 . . 3 ((dom 𝑅𝐴 ∧ ran 𝑆𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
94, 7, 8syl2anc 408 . 2 (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
101, 9sstrid 3108 1 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 3071   × cxp 4537  dom cdm 4539  ran crn 4540   ∘ ccom 4543 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 4046  ax-pow 4098  ax-pr 4131 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 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator