Theorem cossxp2 5291

Description: The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
cossxp2.r	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
cossxp2.s	⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶))

Assertion

Ref	Expression
cossxp2	⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

Proof of Theorem cossxp2

Step	Hyp	Ref	Expression
1		cossxp 5290	. 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2		cossxp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
3		dmxpss2 5200	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴)
4	2, 3	syl 14	. . 3 ⊢ (𝜑 → dom 𝑅 ⊆ 𝐴)
5		cossxp2.s	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶))
6		rnxpss2 5201	. . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶)
8		xpss12 4862	. . 3 ⊢ ((dom 𝑅 ⊆ 𝐴 ∧ ran 𝑆 ⊆ 𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
9	4, 7, 8	syl2anc 411	. 2 ⊢ (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
10	1, 9	sstrid 3253	1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3214   × cxp 4752  dom cdm 4754  ran crn 4755   ∘ ccom 4758

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765

This theorem is referenced by: (None)