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Theorem unirnffid 8413
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)
2 ffn 6185 . . . . 5 (𝐹:𝑇⟶Fin → 𝐹 Fn 𝑇)
31, 2syl 17 . . . 4 (𝜑𝐹 Fn 𝑇)
4 unirnffid.2 . . . 4 (𝜑𝑇 ∈ Fin)
5 fnfi 8393 . . . 4 ((𝐹 Fn 𝑇𝑇 ∈ Fin) → 𝐹 ∈ Fin)
63, 4, 5syl2anc 565 . . 3 (𝜑𝐹 ∈ Fin)
7 rnfi 8404 . . 3 (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
86, 7syl 17 . 2 (𝜑 → ran 𝐹 ∈ Fin)
9 frn 6193 . . 3 (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
101, 9syl 17 . 2 (𝜑 → ran 𝐹 ⊆ Fin)
11 unifi 8410 . 2 ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ran 𝐹 ∈ Fin)
128, 10, 11syl2anc 565 1 (𝜑 ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  wss 3721   cuni 4572  ran crn 5250   Fn wfn 6026  wf 6027  Fincfn 8108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-fin 8112
This theorem is referenced by:  marypha2  8500  acsinfd  17387
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