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Theorem coexg 4889
Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
coexg ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem coexg
StepHypRef Expression
1 cossxp 4870 . 2 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
2 dmexg 4623 . . 3 (𝐵𝑊 → dom 𝐵 ∈ V)
3 rnexg 4624 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
4 xpexg 4479 . . 3 ((dom 𝐵 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐵 × ran 𝐴) ∈ V)
52, 3, 4syl2anr 278 . 2 ((𝐴𝑉𝐵𝑊) → (dom 𝐵 × ran 𝐴) ∈ V)
6 ssexg 3923 . 2 (((𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴𝐵) ∈ V)
71, 5, 6sylancr 399 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  wss 2944   × cxp 4370  dom cdm 4372  ran crn 4373  ccom 4376
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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
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 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383
This theorem is referenced by:  coex  4890
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