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Theorem funexw 7937
Description: Weak version of funex 7220 that holds without ax-rep 5285. 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 7736 . . 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 6565 . . . 4 (Fun 𝐹 → Rel 𝐹)
4 relssdmrn 6267 . . . 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 5324 1 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  Vcvv 3474  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681  Fun wfun 6537
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-fun 6545
This theorem is referenced by:  mptexw  7938  mpoexw  8064  seqexw  13981
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