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Theorem cossxp 5575
 Description: Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
cossxp (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Proof of Theorem cossxp
StepHypRef Expression
1 relco 5550 . . 3 Rel (𝐴𝐵)
2 relssdmrn 5573 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5306 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
5 rncoss 5307 . . 3 ran (𝐴𝐵) ⊆ ran 𝐴
6 xpss12 5148 . . 3 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ ran (𝐴𝐵) ⊆ ran 𝐴) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
74, 5, 6mp2an 704 . 2 (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
83, 7sstri 3577 1 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3540   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049 This theorem is referenced by:  coexg  7010  tposssxp  7243  metustexhalf  22171  rtrclex  36943  trclexi  36946  rtrclexi  36947  cnvtrcl0  36952
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