Theorem cocnvss 5269

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 5268	. 2 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))
2		cossxp 5266	. . 3 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (dom ◡(𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3		df-rn 4742	. . . . 5 ⊢ ran (𝑅 ↾ dom 𝑆) = dom ◡(𝑅 ↾ dom 𝑆)
4	3	eqimss2i 3285	. . . 4 ⊢ dom ◡(𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5		ssid 3248	. . . 4 ⊢ ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6		xpss12 4839	. . . 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 3237	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
9	1, 8	eqsstri 3260	1 ⊢ (𝑆 ∘ ◡𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))

Syntax hints:   ⊆ wss 3201   × cxp 4729  ^◡ccnv 4730  dom cdm 4731  ran crn 4732   ↾ cres 4733   ∘ ccom 4735

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743

This theorem is referenced by:  caserel  7329