Theorem cocnvss 5288

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 5287	. 2 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))
2		cossxp 5285	. . 3 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (dom ◡(𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
3		df-rn 4760	. . . . 5 ⊢ ran (𝑅 ↾ dom 𝑆) = dom ◡(𝑅 ↾ dom 𝑆)
4	3	eqimss2i 3295	. . . 4 ⊢ dom ◡(𝑅 ↾ dom 𝑆) ⊆ ran (𝑅 ↾ dom 𝑆)
5		ssid 3258	. . . 4 ⊢ ran (𝑆 ↾ dom 𝑅) ⊆ ran (𝑆 ↾ dom 𝑅)
6		xpss12 4857	. . . 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 3247	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))
9	1, 8	eqsstri 3270	1 ⊢ (𝑆 ∘ ◡𝑅) ⊆ (ran (𝑅 ↾ dom 𝑆) × ran (𝑆 ↾ dom 𝑅))

Syntax hints:   ⊆ wss 3211   × cxp 4747  ^◡ccnv 4748  dom cdm 4749  ran crn 4750   ↾ cres 4751   ∘ ccom 4753

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761

This theorem is referenced by:  caserel  7378