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Theorem cossxp 5619
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 5594 . . 3 Rel (𝐴𝐵)
2 relssdmrn 5617 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5347 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
5 rncoss 5348 . . 3 ran (𝐴𝐵) ⊆ ran 𝐴
6 xpss12 5188 . . 3 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ ran (𝐴𝐵) ⊆ ran 𝐴) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
74, 5, 6mp2an 707 . 2 (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
83, 7sstri 3593 1 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3556   × cxp 5074  dom cdm 5076  ran crn 5077  ccom 5080  Rel wrel 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087
This theorem is referenced by:  coexg  7067  tposssxp  7304  metustexhalf  22274  rtrclex  37426  trclexi  37429  rtrclexi  37430  cnvtrcl0  37435
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