Theorem cossxp2 5260

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 5259	. 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2		cossxp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
3		dmxpss2 5169	. . . 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 5170	. . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶)
8		xpss12 4833	. . 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 3238	1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3200   × cxp 4723  dom cdm 4725  ran crn 4726   ∘ ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)