Theorem cossxp 5126

Description: Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Assertion

Ref	Expression
cossxp	⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Proof of Theorem cossxp

Step	Hyp	Ref	Expression
1		relco 5102	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 5124	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))
4		dmcoss 4873	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		rncoss 4874	. . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
6		xpss12 4711	. . 3 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
7	4, 5, 6	mp2an 423	. 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
8	3, 7	sstri 3151	1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Syntax hints:   ⊆ wss 3116   × cxp 4602  dom cdm 4604  ran crn 4605   ∘ ccom 4608  Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615

This theorem is referenced by:  cossxp2  5127  cocnvss  5129  coexg  5148  tposssxp  6217