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Theorem f1dmvrnfibi 6578
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6579. (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 5168 . . . 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 5225 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
4 f1ofun 5203 . . . . 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 6576 . . 3  |-  ( ( Rel  F  /\  Fun  `' F  /\  F  e. 
Fin )  ->  ran  F  e.  Fin )
92, 6, 7, 8syl3anc 1170 . 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 5169 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  dom  F  =  A )
12 f1f1orn 5212 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
13 eleq1 2145 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( A  e.  V  <->  dom 
F  e.  V ) )
14 f1oeq2 5193 . . . . . . . . . . . 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 2086 . . . . . . . . . 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 1371 . . . . . . . 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 6404 . . . . 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 6520 . . . 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 5167 . . . . 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 6573 . . . 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 561 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 1285    e. wcel 1434   class class class wbr 3811   `'ccnv 4400   dom cdm 4401   ran crn 4402   Rel wrel 4406   Fun wfun 4963   -1-1->wf1 4966   -1-1-onto->wf1o 4968    ~~ cen 6385   Fincfn 6387
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1st 5846  df-2nd 5847  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390
This theorem is referenced by:  f1vrnfibi  6579
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