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Theorem cocnvss 4943
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 4942 . 2 (𝑆𝑅) = ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆))
2 cossxp 4940 . . 3 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3 df-rn 4439 . . . . 5 ran (𝑅 ↾ dom 𝑆) = dom (𝑅 ↾ dom 𝑆)
43eqimss2i 3079 . . . 4 dom (𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5 ssid 3042 . . . 4 ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6 xpss12 4533 . . . 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 417 . . 3 (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
82, 7sstri 3032 . 2 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
91, 8eqsstri 3054 1 (𝑆𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
Colors of variables: wff set class
Syntax hints:  wss 2997   × cxp 4426  ccnv 4427  dom cdm 4428  ran crn 4429  cres 4430  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440
This theorem is referenced by:  caserel  6757
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