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Theorem cocnvss 5150
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 5149 . 2 (𝑆𝑅) = ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆))
2 cossxp 5147 . . 3 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3 df-rn 4634 . . . . 5 ran (𝑅 ↾ dom 𝑆) = dom (𝑅 ↾ dom 𝑆)
43eqimss2i 3212 . . . 4 dom (𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5 ssid 3175 . . . 4 ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6 xpss12 4730 . . . 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 426 . . 3 (dom (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
82, 7sstri 3164 . 2 ((𝑆 ↾ dom 𝑅) ∘ (𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
91, 8eqsstri 3187 1 (𝑆𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
Colors of variables: wff set class
Syntax hints:  wss 3129   × cxp 4621  ccnv 4622  dom cdm 4623  ran crn 4624  cres 4625  ccom 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635
This theorem is referenced by:  caserel  7080
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