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Theorem funex 5609
 Description: If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5608. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
funex ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)

Proof of Theorem funex
StepHypRef Expression
1 funfn 5121 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnex 5608 . 2 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
31, 2sylanb 280 1 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1463  Vcvv 2658  dom cdm 4507  Fun wfun 5085   Fn wfn 5086 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-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099 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-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  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-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099 This theorem is referenced by:  opabex  5610  mptexg  5611  funrnex  5978  oprabexd  5991  oprabex  5992  mpoexxg  6074
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