Theorem cossxp 6224

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 6063	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 6221	. . 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 5920	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		rncoss 5922	. . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
6		xpss12 5638	. . 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 692	. 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
8	3, 7	sstri 3947	1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Syntax hints:   ⊆ wss 3905   × cxp 5621  dom cdm 5623  ran crn 5624   ∘ ccom 5627  Rel wrel 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634

This theorem is referenced by:  coexg  7869  tposssxp  8170  metustexhalf  24460  rtrclex  43590  trclexi  43593  rtrclexi  43594  cnvtrcl0  43599