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Theorem f1dmvrnfibi 6798
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6799. (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 5300 . . . 4  |-  ( F : A -1-1-> B  ->  Rel  F )
21ad2antlr 478 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Rel  F
)
3 f1cnv 5357 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
4 f1ofun 5335 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  Fun  `' F )
53, 4syl 14 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65ad2antlr 478 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Fun  `' F )
7 simpr 109 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  F  e. 
Fin )
8 funrnfi 6796 . . 3  |-  ( ( Rel  F  /\  Fun  `' F  /\  F  e. 
Fin )  ->  ran  F  e.  Fin )
92, 6, 7, 8syl3anc 1199 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  ran  F  e.  Fin )
10 simpr 109 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  ran  F  e.  Fin )
11 f1dm 5301 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  dom  F  =  A )
12 f1f1orn 5344 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
13 eleq1 2178 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( A  e.  V  <->  dom 
F  e.  V ) )
14 f1oeq2 5325 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( F : A -1-1-onto-> ran  F  <-> 
F : dom  F -1-1-onto-> ran  F ) )
1513, 14anbi12d 462 . . . . . . . . . . 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 2118 . . . . . . . . . 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 143 . . . . . . . . 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 1400 . . . . . . . 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 124 . . . . . 6  |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( dom  F  e.  V  /\  F : dom  F -1-1-onto-> ran  F ) )
2120adantr 272 . . . . 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 6617 . . . . 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 6734 . . . 4  |-  ( ( ran  F  e.  Fin  /\ 
dom  F  ~~  ran  F
)  ->  dom  F  e. 
Fin )
2510, 23, 24syl2anc 406 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  dom  F  e.  Fin )
26 f1fun 5299 . . . . 5  |-  ( F : A -1-1-> B  ->  Fun  F )
2726ad2antlr 478 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  Fun  F )
28 fundmfibi 6793 . . . 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 166 . 2  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  F  e.  Fin )
319, 30impbida 568 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   class class class wbr 3897   `'ccnv 4506   dom cdm 4507   ran crn 4508   Rel wrel 4512   Fun wfun 5085   -1-1->wf1 5088   -1-1-onto->wf1o 5090    ~~ cen 6598   Fincfn 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-er 6395  df-en 6601  df-fin 6603
This theorem is referenced by:  f1vrnfibi  6799
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