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Theorem funrnex 7649
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 6976. (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 6976 . . 3 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
21ex 415 . 2 (Fun 𝐹 → (dom 𝐹𝐵𝐹 ∈ V))
3 rnexg 7608 . 2 (𝐹 ∈ V → ran 𝐹 ∈ V)
42, 3syl6com 37 1 (dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Vcvv 3495  dom cdm 5550  ran crn 5551  Fun wfun 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358
This theorem is referenced by:  zfrep6  7650  fornex  7651  tz7.48-3  8074  inf0  9078  axcc2lem  9852  zorn2lem4  9915  fnct  9953
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