Theorem cossxp 6279

Description: Composition as a subset of the Cartesian 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 6115	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 6276	. . 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 5970	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		rncoss 5972	. . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
6		xpss12 5681	. . 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 705	. 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
8	3, 7	sstri 3949	1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

This proof depends on syntax axioms:   ⊆ wss 3908   × cxp 5664  dom cdm 5666  ran crn 5667   ∘ ccom 5670  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677

This theorem is used by:  coexg  7935  tposssxp  8235  metustexhalf  24750  rtrclex  44384  trclexi  44387  rtrclexi  44388  cnvtrcl0  44393