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Theorem coexg 5051
 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 5029 . 2 (𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
2 dmexg 4771 . . 3 (𝐵𝑊 → dom 𝐵 ∈ V)
3 rnexg 4772 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
4 xpexg 4621 . . 3 ((dom 𝐵 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐵 × ran 𝐴) ∈ V)
52, 3, 4syl2anr 286 . 2 ((𝐴𝑉𝐵𝑊) → (dom 𝐵 × ran 𝐴) ∈ V)
6 ssexg 4035 . 2 (((𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴) ∧ (dom 𝐵 × ran 𝐴) ∈ V) → (𝐴𝐵) ∈ V)
71, 5, 6sylancr 408 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1463  Vcvv 2658   ⊆ wss 3039   × cxp 4505  dom cdm 4507  ran crn 4508   ∘ ccom 4511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518 This theorem is referenced by:  coex  5052  climcncf  12635
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