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Theorem cossxp 4871
Description: Composition as a subset of the cross 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 4847 . . 3 Rel (𝐴𝐵)
2 relssdmrn 4869 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 7 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 4629 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
5 rncoss 4630 . . 3 ran (𝐴𝐵) ⊆ ran 𝐴
6 xpss12 4473 . . 3 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ ran (𝐴𝐵) ⊆ ran 𝐴) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴))
74, 5, 6mp2an 410 . 2 (dom (𝐴𝐵) × ran (𝐴𝐵)) ⊆ (dom 𝐵 × ran 𝐴)
83, 7sstri 2982 1 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wss 2945   × cxp 4371  dom cdm 4373  ran crn 4374  ccom 4377  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384
This theorem is referenced by:  coexg  4890  tposssxp  5895
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