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Theorem funrnex 7004
Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 6365. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
funrnex (dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Proof of Theorem funrnex
StepHypRef Expression
1 funex 6365 . . 3 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
21ex 449 . 2 (Fun 𝐹 → (dom 𝐹𝐵𝐹 ∈ V))
3 rnexg 6968 . 2 (𝐹 ∈ V → ran 𝐹 ∈ V)
42, 3syl6com 36 1 (dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  Vcvv 3173  dom cdm 5028  ran crn 5029  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  zfrep6  7005  fornex  7006  tz7.48-3  7404  inf0  8379  axcc2lem  9119  zorn2lem4  9182  fnct  28670
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