Theorem funrnex 6118

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 5742. (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 5742	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → 𝐹 ∈ V)
2	1	ex 115	. 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ 𝐵 → 𝐹 ∈ V))
3		rnexg 4894	. 2 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
4	2, 3	syl6com 35	1 ⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2148  Vcvv 2739  dom cdm 4628  ran crn 4629  Fun wfun 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226

This theorem is referenced by:  focdmex  6119