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Theorem funexw 7658
 Description: Weak version of funex 6974 that holds without ax-rep 5157. 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 7472 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
213adant1 1128 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
3 funrel 6353 . . . 4 (Fun 𝐹 → Rel 𝐹)
4 relssdmrn 6099 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
53, 4syl 17 . . 3 (Fun 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
653ad2ant1 1131 . 2 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
72, 6ssexd 5195 1 ((Fun 𝐹 ∧ dom 𝐹𝐵 ∧ ran 𝐹𝐶) → 𝐹 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2112  Vcvv 3410   ⊆ wss 3859   × cxp 5523  dom cdm 5525  ran crn 5526  Rel wrel 5530  Fun wfun 6330 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-fun 6338 This theorem is referenced by:  mptexw  7659  mpoexw  7782  seqexw  13435
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