Theorem cossxp2 5070

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 5069	. 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2		cossxp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
3		dmxpss2 4979	. . . 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 4980	. . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶)
8		xpss12 4654	. . 3 ⊢ ((dom 𝑅 ⊆ 𝐴 ∧ ran 𝑆 ⊆ 𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
9	4, 7, 8	syl2anc 409	. 2 ⊢ (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
10	1, 9	sstrid 3113	1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3076   × cxp 4545  dom cdm 4547  ran crn 4548   ∘ ccom 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558

This theorem is referenced by: (None)