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Theorem funex 6523
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 6522. (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 5956 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnex 6522 . 2 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
31, 2sylanb 488 1 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231  dom cdm 5143  Fun wfun 5920   Fn wfn 5921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  opabex  6524  mptexg  6525  mptexgf  6526  funrnex  7175  oprabexd  7197  oprabex  7198  mpt2exxg  7289  tfrlem14  7532  hartogslem2  8489  harwdom  8536  abrexexd  29473  mpt2exxg2  42441
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