Theorem cossxp 6263

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 6101	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 6260	. . 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 5956	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		rncoss 5958	. . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
6		xpss12 5667	. . 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 704	. 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
8	3, 7	sstri 3948	1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Syntax hints:   ⊆ wss 3907   × cxp 5650  dom cdm 5652  ran crn 5653   ∘ ccom 5656  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663

This theorem is referenced by:  coexg  7914  tposssxp  8214  metustexhalf  24674  rtrclex  44205  trclexi  44208  rtrclexi  44209  cnvtrcl0  44214