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Theorem fin4en1 8145
Description: Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin4en1  |-  ( A 
~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )

Proof of Theorem fin4en1
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ensym 7115 . 2  |-  ( A 
~~  B  ->  B  ~~  A )
2 bren 7076 . . . 4  |-  ( B 
~~  A  <->  E. f 
f : B -1-1-onto-> A )
3 simpr 448 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  ->  x  C.  B )
4 f1of1 5632 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  f : B -1-1-> A )
5 pssss 3402 . . . . . . . . . . . . . 14  |-  ( x 
C.  B  ->  x  C_  B )
6 ssid 3327 . . . . . . . . . . . . . 14  |-  B  C_  B
75, 6jctir 525 . . . . . . . . . . . . 13  |-  ( x 
C.  B  ->  (
x  C_  B  /\  B  C_  B ) )
8 f1imapss 5971 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  ( x  C_  B  /\  B  C_  B ) )  ->  ( (
f " x ) 
C.  ( f " B )  <->  x  C.  B ) )
94, 7, 8syl2an 464 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
x  C.  B )
)
103, 9mpbird 224 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  ( f " B ) )
11 imadmrn 5174 . . . . . . . . . . . . . 14  |-  ( f
" dom  f )  =  ran  f
12 f1odm 5637 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  ->  dom  f  =  B )
1312imaeq2d 5162 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ( f " dom  f )  =  ( f " B
) )
14 dff1o5 5642 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  <->  ( f : B -1-1-> A  /\  ran  f  =  A ) )
1514simprbi 451 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ran  f  =  A )
1611, 13, 153eqtr3a 2460 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  ( f " B )  =  A )
1716adantr 452 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " B
)  =  A )
1817psseq2d 3400 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
( f " x
)  C.  A )
)
1910, 18mpbid 202 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  A )
2019adantrr 698 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  C.  A
)
21 vex 2919 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221f1imaen 7128 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  x  C_  B )  ->  ( f "
x )  ~~  x
)
234, 5, 22syl2an 464 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  ~~  x )
2423adantrr 698 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  x
)
25 simprr 734 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
26 entr 7118 . . . . . . . . . . 11  |-  ( ( ( f " x
)  ~~  x  /\  x  ~~  B )  -> 
( f " x
)  ~~  B )
2724, 25, 26syl2anc 643 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  B
)
28 vex 2919 . . . . . . . . . . . 12  |-  f  e. 
_V
29 f1oen3g 7082 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : B -1-1-onto-> A )  ->  B  ~~  A )
3028, 29mpan 652 . . . . . . . . . . 11  |-  ( f : B -1-1-onto-> A  ->  B  ~~  A )
3130adantr 452 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  B  ~~  A
)
32 entr 7118 . . . . . . . . . 10  |-  ( ( ( f " x
)  ~~  B  /\  B  ~~  A )  -> 
( f " x
)  ~~  A )
3327, 31, 32syl2anc 643 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  A
)
34 fin4i 8134 . . . . . . . . 9  |-  ( ( ( f " x
)  C.  A  /\  ( f " x
)  ~~  A )  ->  -.  A  e. FinIV )
3520, 33, 34syl2anc 643 . . . . . . . 8  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
3635ex 424 . . . . . . 7  |-  ( f : B -1-1-onto-> A  ->  ( (
x  C.  B  /\  x  ~~  B )  ->  -.  A  e. FinIV ) )
3736exlimdv 1643 . . . . . 6  |-  ( f : B -1-1-onto-> A  ->  ( E. x ( x  C.  B  /\  x  ~~  B
)  ->  -.  A  e. FinIV
) )
3837con2d 109 . . . . 5  |-  ( f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
3938exlimiv 1641 . . . 4  |-  ( E. f  f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B
) ) )
402, 39sylbi 188 . . 3  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  -.  E. x
( x  C.  B  /\  x  ~~  B ) ) )
41 relen 7073 . . . . 5  |-  Rel  ~~
4241brrelexi 4877 . . . 4  |-  ( B 
~~  A  ->  B  e.  _V )
43 isfin4 8133 . . . 4  |-  ( B  e.  _V  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4442, 43syl 16 . . 3  |-  ( B 
~~  A  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4540, 44sylibrd 226 . 2  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
461, 45syl 16 1  |-  ( A 
~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280    C. wpss 3281   class class class wbr 4172   dom cdm 4837   ran crn 4838   "cima 4840   -1-1->wf1 5410   -1-1-onto->wf1o 5412    ~~ cen 7065  FinIVcfin4 8116
This theorem is referenced by:  domfin4  8147  isfin4-3  8151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-fin4 8123
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