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Theorem unirnffid 8809
 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 6504 . . . 4 (𝜑𝐹 Fn 𝑇)
3 unirnffid.2 . . . 4 (𝜑𝑇 ∈ Fin)
4 fnfi 8789 . . . 4 ((𝐹 Fn 𝑇𝑇 ∈ Fin) → 𝐹 ∈ Fin)
52, 3, 4syl2anc 587 . . 3 (𝜑𝐹 ∈ Fin)
6 rnfi 8800 . . 3 (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
75, 6syl 17 . 2 (𝜑 → ran 𝐹 ∈ Fin)
81frnd 6510 . 2 (𝜑 → ran 𝐹 ⊆ Fin)
9 unifi 8806 . 2 ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ran 𝐹 ∈ Fin)
107, 8, 9syl2anc 587 1 (𝜑 ran 𝐹 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115   ⊆ wss 3919  ∪ cuni 4825  ran crn 5544   Fn wfn 6339  ⟶wf 6340  Fincfn 8501 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-fin 8505 This theorem is referenced by:  marypha2  8896  acsinfd  17788
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