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Theorem fin4en1 8757
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 7636 . 2  |-  ( A 
~~  B  ->  B  ~~  A )
2 bren 7596 . . . 4  |-  ( B 
~~  A  <->  E. f 
f : B -1-1-onto-> A )
3 simpr 468 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  ->  x  C.  B )
4 f1of1 5827 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  f : B -1-1-> A )
5 pssss 3514 . . . . . . . . . . . . . 14  |-  ( x 
C.  B  ->  x  C_  B )
6 ssid 3437 . . . . . . . . . . . . . 14  |-  B  C_  B
75, 6jctir 547 . . . . . . . . . . . . 13  |-  ( x 
C.  B  ->  (
x  C_  B  /\  B  C_  B ) )
8 f1imapss 6185 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  ( x  C_  B  /\  B  C_  B ) )  ->  ( (
f " x ) 
C.  ( f " B )  <->  x  C.  B
) )
94, 7, 8syl2an 485 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
x  C.  B )
)
103, 9mpbird 240 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  ( f " B ) )
11 imadmrn 5184 . . . . . . . . . . . . . 14  |-  ( f
" dom  f )  =  ran  f
12 f1odm 5832 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  ->  dom  f  =  B )
1312imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ( f " dom  f )  =  ( f " B
) )
14 dff1o5 5837 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  <->  ( f : B -1-1-> A  /\  ran  f  =  A ) )
1514simprbi 471 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ran  f  =  A )
1611, 13, 153eqtr3a 2529 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  ( f " B )  =  A )
1716adantr 472 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " B
)  =  A )
1817psseq2d 3512 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
( f " x
)  C.  A )
)
1910, 18mpbid 215 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  A )
2019adantrr 731 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  C.  A
)
21 vex 3034 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221f1imaen 7649 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  x  C_  B )  ->  ( f "
x )  ~~  x
)
234, 5, 22syl2an 485 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  ~~  x )
2423adantrr 731 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  x
)
25 simprr 774 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
26 entr 7639 . . . . . . . . . . 11  |-  ( ( ( f " x
)  ~~  x  /\  x  ~~  B )  -> 
( f " x
)  ~~  B )
2724, 25, 26syl2anc 673 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  B
)
28 vex 3034 . . . . . . . . . . . 12  |-  f  e. 
_V
29 f1oen3g 7603 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : B -1-1-onto-> A )  ->  B  ~~  A )
3028, 29mpan 684 . . . . . . . . . . 11  |-  ( f : B -1-1-onto-> A  ->  B  ~~  A )
3130adantr 472 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  B  ~~  A
)
32 entr 7639 . . . . . . . . . 10  |-  ( ( ( f " x
)  ~~  B  /\  B  ~~  A )  -> 
( f " x
)  ~~  A )
3327, 31, 32syl2anc 673 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  A
)
34 fin4i 8746 . . . . . . . . 9  |-  ( ( ( f " x
)  C.  A  /\  ( f " x
)  ~~  A )  ->  -.  A  e. FinIV )
3520, 33, 34syl2anc 673 . . . . . . . 8  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
3635ex 441 . . . . . . 7  |-  ( f : B -1-1-onto-> A  ->  ( (
x  C.  B  /\  x  ~~  B )  ->  -.  A  e. FinIV ) )
3736exlimdv 1787 . . . . . 6  |-  ( f : B -1-1-onto-> A  ->  ( E. x ( x  C.  B  /\  x  ~~  B
)  ->  -.  A  e. FinIV
) )
3837con2d 119 . . . . 5  |-  ( f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
3938exlimiv 1784 . . . 4  |-  ( E. f  f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B
) ) )
402, 39sylbi 200 . . 3  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  -.  E. x
( x  C.  B  /\  x  ~~  B ) ) )
41 relen 7592 . . . . 5  |-  Rel  ~~
4241brrelexi 4880 . . . 4  |-  ( B 
~~  A  ->  B  e.  _V )
43 isfin4 8745 . . . 4  |-  ( B  e.  _V  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4442, 43syl 17 . . 3  |-  ( B 
~~  A  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4540, 44sylibrd 242 . 2  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
461, 45syl 17 1  |-  ( A 
~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031    C_ wss 3390    C. wpss 3391   class class class wbr 4395   dom cdm 4839   ran crn 4840   "cima 4842   -1-1->wf1 5586   -1-1-onto->wf1o 5588    ~~ cen 7584  FinIVcfin4 8728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-fin4 8735
This theorem is referenced by:  domfin4  8759  isfin4-3  8763
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