Theorem funrnex 7770

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 7077. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
funrnex	⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Proof of Theorem funrnex

Step	Hyp	Ref	Expression
1		funex 7077	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V)
2	1	ex 412	. 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ 𝐵 → 𝐹 ∈ V))
3		rnexg 7725	. 2 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
4	2, 3	syl6com 37	1 ⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422  dom cdm 5580  ran crn 5581  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  zfrep6  7771  fornex  7772  tz7.48-3  8245  inf0  9309  axcc2lem  10123  zorn2lem4  10186  fnct  10224