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Theorem funex 6201
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 6200. (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 5662 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnex 6200 . 2 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
31, 2sylanb 482 1 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 378  wcel 1937  Vcvv 3066  dom cdm 4880  Fun wfun 5627   Fn wfn 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641
This theorem is referenced by:  opabex  6202  mptexg  6203  funrnex  6837  oprabexd  6857  oprabex  6858  mpt2exxg  6944  tfrlem14  7186  hartogslem2  8141  harwdom  8188  abrexexd  28304  mptexgf  28383  mpt2exxg2  41309
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