Theorem cocnvss 5253

Description: Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
cocnvss	⊢ (𝑆 ∘ ◡𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))

Proof of Theorem cocnvss

Step	Hyp	Ref	Expression
1		cocnvres 5252	. 2 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))
2		cossxp 5250	. . 3 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (dom ◡(𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3		df-rn 4729	. . . . 5 ⊢ ran (𝑅 ↾ dom 𝑆) = dom ◡(𝑅 ↾ dom 𝑆)
4	3	eqimss2i 3281	. . . 4 ⊢ dom ◡(𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5		ssid 3244	. . . 4 ⊢ ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6		xpss12 4825	. . . 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)))
7	4, 5, 6	mp2an 426	. . 3 ⊢ (dom ◡(𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
8	2, 7	sstri 3233	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
9	1, 8	eqsstri 3256	1 ⊢ (𝑆 ∘ ◡𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))

Syntax hints:   ⊆ wss 3197   × cxp 4716  ^◡ccnv 4717  dom cdm 4718  ran crn 4719   ↾ cres 4720   ∘ ccom 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730

This theorem is referenced by:  caserel  7250