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Theorem f1dmvrnfibi 6942
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6943. (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 5425 . . . 4  |-  ( F : A -1-1-> B  ->  Rel  F )
21ad2antlr 489 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Rel  F
)
3 f1cnv 5485 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
4 f1ofun 5463 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  Fun  `' F )
53, 4syl 14 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65ad2antlr 489 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Fun  `' F )
7 simpr 110 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  F  e. 
Fin )
8 funrnfi 6940 . . 3  |-  ( ( Rel  F  /\  Fun  `' F  /\  F  e. 
Fin )  ->  ran  F  e.  Fin )
92, 6, 7, 8syl3anc 1238 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  ran  F  e.  Fin )
10 simpr 110 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  ran  F  e.  Fin )
11 f1dm 5426 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  dom  F  =  A )
12 f1f1orn 5472 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
13 eleq1 2240 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( A  e.  V  <->  dom 
F  e.  V ) )
14 f1oeq2 5450 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( F : A -1-1-onto-> ran  F  <-> 
F : dom  F -1-1-onto-> ran  F ) )
1513, 14anbi12d 473 . . . . . . . . . . 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 2180 . . . . . . . . . 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 144 . . . . . . . . 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 1441 . . . . . . . 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 62 . . . . . . 7  |-  ( F : A -1-1-> B  -> 
( A  e.  V  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) ) )
2019impcom 125 . . . . . 6  |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) )
2120adantr 276 . . . . 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 6756 . . . . 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 6873 . . . 4  |-  ( ( ran  F  e.  Fin  /\ 
dom  F  ~~  ran  F
)  ->  dom  F  e. 
Fin )
2510, 23, 24syl2anc 411 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  dom  F  e.  Fin )
26 f1fun 5424 . . . . 5  |-  ( F : A -1-1-> B  ->  Fun  F )
2726ad2antlr 489 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  Fun  F )
28 fundmfibi 6937 . . . 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 167 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  F  e.  Fin )
319, 30impbida 596 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4003   `'ccnv 4625   dom cdm 4626   ran crn 4627   Rel wrel 4631   Fun wfun 5210   -1-1->wf1 5213   -1-1-onto->wf1o 5215    ~~ cen 6737   Fincfn 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-er 6534  df-en 6740  df-fin 6742
This theorem is referenced by:  f1vrnfibi  6943
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