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Theorem funexw 7920
Description: Weak version of funex 7205 that holds without ax-rep 5278. 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 7720 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
213adant1 1130 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
3 funrel 6554 . . . 4 (Fun 𝐹 → Rel 𝐹)
4 relssdmrn 6256 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
53, 4syl 17 . . 3 (Fun 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
653ad2ant1 1133 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
72, 6ssexd 5317 1 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  Vcvv 3473  wss 3944   × cxp 5667  dom cdm 5669  ran crn 5670  Rel wrel 5674  Fun wfun 6526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-fun 6534
This theorem is referenced by:  mptexw  7921  mpoexw  8047  seqexw  13964
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