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Theorem funexw 7932
Description: Weak version of funex 7213 that holds without ax-rep 5276. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Assertion
Ref Expression
funexw ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)

Proof of Theorem funexw
StepHypRef Expression
1 xpexg 7731 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
213adant1 1127 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
3 funrel 6556 . . . 4 (Fun 𝐹 → Rel 𝐹)
4 relssdmrn 6258 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
53, 4syl 17 . . 3 (Fun 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
653ad2ant1 1130 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
72, 6ssexd 5315 1 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  Vcvv 3466  wss 3941   × cxp 5665  dom cdm 5667  ran crn 5668  Rel wrel 5672  Fun wfun 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-fun 6536
This theorem is referenced by:  mptexw  7933  mpoexw  8059  seqexw  13983
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