Theorem funex 5327

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 5326. (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 𝐹 ∈ B) → 𝐹 ∈ V)

Proof of Theorem funex

Step	Hyp	Ref	Expression
1		funfn 4874	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fnex 5326	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ B) → 𝐹 ∈ V)
3	1, 2	sylanb 268	1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ B) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551  dom cdm 4288  Fun wfun 4839   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  opabex  5328  mptexg  5329  funrnex  5683  oprabexd  5696  oprabex  5697  mpt2exxg  5775