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Theorem cossxp2 5111
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 5110 . 2 (𝑆𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2 cossxp2.r . . . 4 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
3 dmxpss2 5020 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅𝐴)
42, 3syl 14 . . 3 (𝜑 → dom 𝑅𝐴)
5 cossxp2.s . . . 4 (𝜑𝑆 ⊆ (𝐵 × 𝐶))
6 rnxpss2 5021 . . . 4 (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆𝐶)
75, 6syl 14 . . 3 (𝜑 → ran 𝑆𝐶)
8 xpss12 4695 . . 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 3139 1 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3102   × cxp 4586  dom cdm 4588  ran crn 4589  ccom 4592
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599
This theorem is referenced by: (None)
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