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Theorem funex 6364
Description: If the domain of a function exists, so 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 6363. (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 5818 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnex 6363 . 2 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
31, 2sylanb 487 1 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  Vcvv 3171  dom cdm 5027  Fun wfun 5783   Fn wfn 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2588  ax-rep 4692  ax-sep 4702  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2460  df-mo 2461  df-clab 2595  df-cleq 2601  df-clel 2604  df-nfc 2738  df-ne 2780  df-ral 2899  df-rex 2900  df-reu 2901  df-rab 2903  df-v 3173  df-sbc 3401  df-csb 3498  df-dif 3541  df-un 3543  df-in 3545  df-ss 3552  df-nul 3873  df-if 4035  df-sn 4124  df-pr 4126  df-op 4130  df-uni 4366  df-iun 4450  df-br 4577  df-opab 4637  df-mpt 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797
This theorem is referenced by:  opabex  6365  mptexg  6366  funrnex  7002  oprabexd  7022  oprabex  7023  mpt2exxg  7109  tfrlem14  7350  hartogslem2  8307  harwdom  8354  abrexexd  28536  mptexgf  28614  mpt2exxg2  41910
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