MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unirnffid Structured version   Unicode version

Theorem unirnffid 7398
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  |-  ( ph  ->  F : T --> Fin )
unirnffid.2  |-  ( ph  ->  T  e.  Fin )
Assertion
Ref Expression
unirnffid  |-  ( ph  ->  U. ran  F  e. 
Fin )

Proof of Theorem unirnffid
StepHypRef Expression
1 unirnffid.1 . . . . 5  |-  ( ph  ->  F : T --> Fin )
2 ffn 5591 . . . . 5  |-  ( F : T --> Fin  ->  F  Fn  T )
31, 2syl 16 . . . 4  |-  ( ph  ->  F  Fn  T )
4 unirnffid.2 . . . 4  |-  ( ph  ->  T  e.  Fin )
5 fnfi 7384 . . . 4  |-  ( ( F  Fn  T  /\  T  e.  Fin )  ->  F  e.  Fin )
63, 4, 5syl2anc 643 . . 3  |-  ( ph  ->  F  e.  Fin )
7 rnfi 7391 . . 3  |-  ( F  e.  Fin  ->  ran  F  e.  Fin )
86, 7syl 16 . 2  |-  ( ph  ->  ran  F  e.  Fin )
9 frn 5597 . . 3  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
101, 9syl 16 . 2  |-  ( ph  ->  ran  F  C_  Fin )
11 unifi 7395 . 2  |-  ( ( ran  F  e.  Fin  /\ 
ran  F  C_  Fin )  ->  U. ran  F  e. 
Fin )
128, 10, 11syl2anc 643 1  |-  ( ph  ->  U. ran  F  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3320   U.cuni 4015   ran crn 4879    Fn wfn 5449   -->wf 5450   Fincfn 7109
This theorem is referenced by:  marypha2  7444  acsinfd  14606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-fin 7113
  Copyright terms: Public domain W3C validator