Theorem cossxp2 5314

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 5313	. 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆)
2		cossxp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
3		dmxpss2 5223	. . . 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 5224	. . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶)
8		xpss12 4884	. . 3 ⊢ ((dom 𝑅 ⊆ 𝐴 ∧ ran 𝑆 ⊆ 𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
9	4, 7, 8	syl2anc 415	. 2 ⊢ (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶))
10	1, 9	sstrid 3259	1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3220   × cxp 4774  dom cdm 4776  ran crn 4777   ∘ ccom 4780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787

This theorem is used by: (None)