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Theorem cossxp 5819
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 5794 . . 3 Rel (𝐴𝐵)
2 relssdmrn 5817 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5540 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
5 rncoss 5541 . . 3 ran (𝐴𝐵) ⊆ ran 𝐴
6 xpss12 5281 . . 3 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ ran (𝐴𝐵) ⊆ ran 𝐴) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
74, 5, 6mp2an 710 . 2 (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
83, 7sstri 3753 1 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3715   × cxp 5264  dom cdm 5266  ran crn 5267  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277
This theorem is referenced by:  coexg  7283  tposssxp  7526  metustexhalf  22582  rtrclex  38444  trclexi  38447  rtrclexi  38448  cnvtrcl0  38453
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