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Theorem unirnffid 9072
Description: The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
unirnffid.1 (𝜑𝐹:𝑇⟶Fin)
unirnffid.2 (𝜑𝑇 ∈ Fin)
Assertion
Ref Expression
unirnffid (𝜑 ran 𝐹 ∈ Fin)

Proof of Theorem unirnffid
StepHypRef Expression
1 unirnffid.1 . . . . 5 (𝜑𝐹:𝑇⟶Fin)
21ffnd 6597 . . . 4 (𝜑𝐹 Fn 𝑇)
3 unirnffid.2 . . . 4 (𝜑𝑇 ∈ Fin)
4 fnfi 8929 . . . 4 ((𝐹 Fn 𝑇𝑇 ∈ Fin) → 𝐹 ∈ Fin)
52, 3, 4syl2anc 583 . . 3 (𝜑𝐹 ∈ Fin)
6 rnfi 9063 . . 3 (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
75, 6syl 17 . 2 (𝜑 → ran 𝐹 ∈ Fin)
81frnd 6604 . 2 (𝜑 → ran 𝐹 ⊆ Fin)
9 unifi 9069 . 2 ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ran 𝐹 ∈ Fin)
107, 8, 9syl2anc 583 1 (𝜑 ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  wss 3891   cuni 4844  ran crn 5589   Fn wfn 6425  wf 6426  Fincfn 8707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1st 7817  df-2nd 7818  df-1o 8281  df-er 8472  df-en 8708  df-dom 8709  df-fin 8711
This theorem is referenced by:  marypha2  9159  acsinfd  18255
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