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Theorem relfi 28052
Description: A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
Assertion
Ref Expression
relfi  |-  ( Rel 
A  ->  ( A  e.  Fin  <->  ( dom  A  e.  Fin  /\  ran  A  e.  Fin ) ) )

Proof of Theorem relfi
StepHypRef Expression
1 dmfi 7860 . . 3  |-  ( A  e.  Fin  ->  dom  A  e.  Fin )
2 rnfi 7863 . . 3  |-  ( A  e.  Fin  ->  ran  A  e.  Fin )
31, 2jca 534 . 2  |-  ( A  e.  Fin  ->  ( dom  A  e.  Fin  /\  ran  A  e.  Fin )
)
4 xpfi 7848 . . . 4  |-  ( ( dom  A  e.  Fin  /\ 
ran  A  e.  Fin )  ->  ( dom  A  X.  ran  A )  e. 
Fin )
5 relssdmrn 5376 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
6 ssfi 7798 . . . 4  |-  ( ( ( dom  A  X.  ran  A )  e.  Fin  /\  A  C_  ( dom  A  X.  ran  A ) )  ->  A  e.  Fin )
74, 5, 6syl2anr 480 . . 3  |-  ( ( Rel  A  /\  ( dom  A  e.  Fin  /\  ran  A  e.  Fin )
)  ->  A  e.  Fin )
87ex 435 . 2  |-  ( Rel 
A  ->  ( ( dom  A  e.  Fin  /\  ran  A  e.  Fin )  ->  A  e.  Fin )
)
93, 8impbid2 207 1  |-  ( Rel 
A  ->  ( A  e.  Fin  <->  ( dom  A  e.  Fin  /\  ran  A  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1870    C_ wss 3442    X. cxp 4852   dom cdm 4854   ran crn 4855   Rel wrel 4859   Fincfn 7577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-fin 7581
This theorem is referenced by:  fpwrelmapffslem  28160
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