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Theorem f1dmvrnfibi 6800
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6801. (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 5302 . . . 4  |-  ( F : A -1-1-> B  ->  Rel  F )
21ad2antlr 480 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  F  e.  Fin )  ->  Rel  F
)
3 f1cnv 5359 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
4 f1ofun 5337 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  Fun  `' F )
53, 4syl 14 . . . 4  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65ad2antlr 480 . . 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 6798 . . 3  |-  ( ( Rel  F  /\  Fun  `' F  /\  F  e. 
Fin )  ->  ran  F  e.  Fin )
92, 6, 7, 8syl3anc 1201 . 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 5303 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  dom  F  =  A )
12 f1f1orn 5346 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
13 eleq1 2180 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( A  e.  V  <->  dom 
F  e.  V ) )
14 f1oeq2 5327 . . . . . . . . . . . 12  |-  ( A  =  dom  F  -> 
( F : A -1-1-onto-> ran  F  <-> 
F : dom  F -1-1-onto-> ran  F ) )
1513, 14anbi12d 464 . . . . . . . . . . 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 2120 . . . . . . . . . 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 1402 . . . . . . . 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 274 . . . . 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 6619 . . . . 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 6736 . . . 4  |-  ( ( ran  F  e.  Fin  /\ 
dom  F  ~~  ran  F
)  ->  dom  F  e. 
Fin )
2510, 23, 24syl2anc 408 . . 3  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  dom  F  e.  Fin )
26 f1fun 5301 . . . . 5  |-  ( F : A -1-1-> B  ->  Fun  F )
2726ad2antlr 480 . . . 4  |-  ( ( ( A  e.  V  /\  F : A -1-1-> B
)  /\  ran  F  e. 
Fin )  ->  Fun  F )
28 fundmfibi 6795 . . . 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 570 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 1316    e. wcel 1465   class class class wbr 3899   `'ccnv 4508   dom cdm 4509   ran crn 4510   Rel wrel 4514   Fun wfun 5087   -1-1->wf1 5090   -1-1-onto->wf1o 5092    ~~ cen 6600   Fincfn 6602
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by:  f1vrnfibi  6801
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