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Theorem funexw 7885
Description: Weak version of funex 7170 that holds without ax-rep 5243. 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 7685 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
213adant1 1131 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
3 funrel 6519 . . . 4 (Fun 𝐹 → Rel 𝐹)
4 relssdmrn 6221 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
53, 4syl 17 . . 3 (Fun 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
653ad2ant1 1134 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
72, 6ssexd 5282 1 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wcel 2107  Vcvv 3444  wss 3911   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-fun 6499
This theorem is referenced by:  mptexw  7886  mpoexw  8012  seqexw  13928
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