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Theorem f1dmvrnfibi 6651
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6652. (Contributed by AV, 10-Jan-2020.)
Assertion
Ref Expression
f1dmvrnfibi  |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( F  e. 
Fin 
<->  ran  F  e.  Fin ) )

Proof of Theorem f1dmvrnfibi
StepHypRef Expression
1 f1rel 5220 . . . 4  |-  ( F : A -1-1-> B  ->  Rel  F )
21ad2antlr 473 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Rel  F
)
3 f1cnv 5277 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
4 f1ofun 5255 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  Fun  `' F )
53, 4syl 14 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65ad2antlr 473 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Fun  `' F )
7 simpr 108 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  F  e. 
Fin )
8 funrnfi 6649 . . 3  |-  ( ( Rel  F  /\  Fun  `' F  /\  F  e. 
Fin )  ->  ran  F  e.  Fin )
92, 6, 7, 8syl3anc 1174 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  ran  F  e.  Fin )
10 simpr 108 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  ran  F  e.  Fin )
11 f1dm 5221 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  dom  F  =  A )
12 f1f1orn 5264 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
13 eleq1 2150 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( A  e.  V  <->  dom 
F  e.  V ) )
14 f1oeq2 5245 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( F : A -1-1-onto-> ran  F  <-> 
F : dom  F -1-1-onto-> ran  F ) )
1513, 14anbi12d 457 . . . . . . . . . . 11  |-  ( A  =  dom  F  -> 
( ( A  e.  V  /\  F : A
-1-1-onto-> ran  F )  <->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) ) )
1615eqcoms 2091 . . . . . . . . . 10  |-  ( dom 
F  =  A  -> 
( ( A  e.  V  /\  F : A
-1-1-onto-> ran  F )  <->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) ) )
1716biimpd 142 . . . . . . . . 9  |-  ( dom 
F  =  A  -> 
( ( A  e.  V  /\  F : A
-1-1-onto-> ran  F )  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F
) ) )
1817expcomd 1375 . . . . . . . 8  |-  ( dom 
F  =  A  -> 
( F : A -1-1-onto-> ran  F  ->  ( A  e.  V  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) ) ) )
1911, 12, 18sylc 61 . . . . . . 7  |-  ( F : A -1-1-> B  -> 
( A  e.  V  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) ) )
2019impcom 123 . . . . . 6  |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) )
2120adantr 270 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F
) )
22 f1oeng 6472 . . . . 5  |-  ( ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F )  ->  dom  F  ~~  ran  F )
2321, 22syl 14 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  dom  F 
~~  ran  F )
24 enfii 6588 . . . 4  |-  ( ( ran  F  e.  Fin  /\ 
dom  F  ~~  ran  F
)  ->  dom  F  e. 
Fin )
2510, 23, 24syl2anc 403 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  dom  F  e.  Fin )
26 f1fun 5219 . . . . 5  |-  ( F : A -1-1-> B  ->  Fun  F )
2726ad2antlr 473 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  Fun  F )
28 fundmfibi 6646 . . . 4  |-  ( Fun 
F  ->  ( F  e.  Fin  <->  dom  F  e.  Fin ) )
2927, 28syl 14 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  ( F  e.  Fin  <->  dom  F  e. 
Fin ) )
3025, 29mpbird 165 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  F  e.  Fin )
319, 30impbida 563 1  |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( F  e. 
Fin 
<->  ran  F  e.  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443   Fun wfun 5009   -1-1->wf1 5012   -1-1-onto->wf1o 5014    ~~ cen 6453   Fincfn 6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-er 6290  df-en 6456  df-fin 6458
This theorem is referenced by:  f1vrnfibi  6652
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