Theorem funex 7174

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 7172. (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 6528	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fnex 7172	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V)
3	1, 2	sylanb 582	1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3429  dom cdm 5631  Fun wfun 6492   Fn wfn 6493

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by:  opabex  7175  mptexg  7176  mptexgf  7177  funrnex  7907  oprabexd  7928  oprabex  7929  mpoexxg  8028  tfrlem14  8330  hartogslem2  9458  harwdom  9506  abrexexd  32579  modelaxreplem2  45406  mpoexxg2  48814