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Theorem cocnvss 5064
 Description: Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
cocnvss (𝑆𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))

Proof of Theorem cocnvss
StepHypRef Expression
1 cocnvres 5063 . 2 (𝑆𝑅) = ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆))
2 cossxp 5061 . . 3 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3 df-rn 4550 . . . . 5 ran (𝑅 ↾ dom 𝑆) = dom (𝑅 ↾ dom 𝑆)
43eqimss2i 3154 . . . 4 dom (𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5 ssid 3117 . . . 4 ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6 xpss12 4646 . . . 4 ((dom (𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆) ∧ ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)) → (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)))
74, 5, 6mp2an 422 . . 3 (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
82, 7sstri 3106 . 2 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
91, 8eqsstri 3129 1 (𝑆𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
 Colors of variables: wff set class Syntax hints:   ⊆ wss 3071   × cxp 4537  ◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   ∘ 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  df-res 4551 This theorem is referenced by:  caserel  6972
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