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Theorem cossxp 6293
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 6128 . . 3 Rel (𝐴𝐵)
2 relssdmrn 6289 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5987 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
5 rncoss 5988 . . 3 ran (𝐴𝐵) ⊆ ran 𝐴
6 xpss12 5703 . . 3 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ ran (𝐴𝐵) ⊆ ran 𝐴) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
74, 5, 6mp2an 692 . 2 (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
83, 7sstri 4004 1 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3962   × cxp 5686  dom cdm 5688  ran crn 5689  ccom 5692  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699
This theorem is referenced by:  coexg  7951  tposssxp  8253  metustexhalf  24584  rtrclex  43606  trclexi  43609  rtrclexi  43610  cnvtrcl0  43615
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