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Theorem funex 6963
 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 6961. (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 6358 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnex 6961 . 2 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
31, 2sylanb 584 1 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  Vcvv 3444  dom cdm 5523  Fun wfun 6322   Fn wfn 6323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336 This theorem is referenced by:  opabex  6964  mptexg  6965  mptexgf  6966  funrnex  7641  oprabexd  7662  oprabex  7663  mpoexxg  7760  tfrlem14  8014  hartogslem2  8995  harwdom  9043  abrexexd  30280  mpoexxg2  44726
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