Theorem cossxp2 5170

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 5169	. 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2		cossxp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
3		dmxpss2 5079	. . . 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 5080	. . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶)
8		xpss12 4751	. . 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 3181	1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3144   × cxp 4642  dom cdm 4644  ran crn 4645   ∘ ccom 4648

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655

This theorem is referenced by: (None)