Theorem cossxp 5261

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 5237	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 5259	. . 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 5004	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		rncoss 5005	. . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
6		xpss12 4835	. . 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 426	. 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
8	3, 7	sstri 3235	1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Syntax hints:   ⊆ wss 3199   × cxp 4725  dom cdm 4727  ran crn 4728   ∘ ccom 4731  Rel wrel 4732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738

This theorem is referenced by:  cossxp2  5262  cocnvss  5264  coexg  5283  tposssxp  6420