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Theorem cossxp2 5134
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 5133 . 2 (𝑆𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2 cossxp2.r . . . 4 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
3 dmxpss2 5043 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
42, 3syl 14 . . 3 (𝜑 → dom 𝑅𝐴)
5 cossxp2.s . . . 4 (𝜑𝑆 ⊆ (𝐵 × 𝐶))
6 rnxpss2 5044 . . . 4 (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆𝐶)
75, 6syl 14 . . 3 (𝜑 → ran 𝑆𝐶)
8 xpss12 4718 . . 3 ((dom 𝑅𝐴 ∧ ran 𝑆𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
94, 7, 8syl2anc 409 . 2 (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
101, 9sstrid 3158 1 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3121   × cxp 4609  dom cdm 4611  ran crn 4612  ccom 4615
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622
This theorem is referenced by: (None)
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