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Theorem fin4en1 8685
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 7561 . 2  |-  ( A 
~~  B  ->  B  ~~  A )
2 bren 7522 . . . 4  |-  ( B 
~~  A  <->  E. f 
f : B -1-1-onto-> A )
3 simpr 461 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  ->  x  C.  B )
4 f1of1 5813 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  f : B -1-1-> A )
5 pssss 3599 . . . . . . . . . . . . . 14  |-  ( x 
C.  B  ->  x  C_  B )
6 ssid 3523 . . . . . . . . . . . . . 14  |-  B  C_  B
75, 6jctir 538 . . . . . . . . . . . . 13  |-  ( x 
C.  B  ->  (
x  C_  B  /\  B  C_  B ) )
8 f1imapss 6160 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  ( x  C_  B  /\  B  C_  B ) )  ->  ( (
f " x ) 
C.  ( f " B )  <->  x  C.  B
) )
94, 7, 8syl2an 477 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
x  C.  B )
)
103, 9mpbird 232 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  ( f " B ) )
11 imadmrn 5345 . . . . . . . . . . . . . 14  |-  ( f
" dom  f )  =  ran  f
12 f1odm 5818 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  ->  dom  f  =  B )
1312imaeq2d 5335 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ( f " dom  f )  =  ( f " B
) )
14 dff1o5 5823 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  <->  ( f : B -1-1-> A  /\  ran  f  =  A ) )
1514simprbi 464 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ran  f  =  A )
1611, 13, 153eqtr3a 2532 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  ( f " B )  =  A )
1716adantr 465 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " B
)  =  A )
1817psseq2d 3597 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
( f " x
)  C.  A )
)
1910, 18mpbid 210 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  A )
2019adantrr 716 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  C.  A
)
21 vex 3116 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221f1imaen 7574 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  x  C_  B )  ->  ( f "
x )  ~~  x
)
234, 5, 22syl2an 477 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  ~~  x )
2423adantrr 716 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  x
)
25 simprr 756 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
26 entr 7564 . . . . . . . . . . 11  |-  ( ( ( f " x
)  ~~  x  /\  x  ~~  B )  -> 
( f " x
)  ~~  B )
2724, 25, 26syl2anc 661 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  B
)
28 vex 3116 . . . . . . . . . . . 12  |-  f  e. 
_V
29 f1oen3g 7528 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : B -1-1-onto-> A )  ->  B  ~~  A )
3028, 29mpan 670 . . . . . . . . . . 11  |-  ( f : B -1-1-onto-> A  ->  B  ~~  A )
3130adantr 465 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  B  ~~  A
)
32 entr 7564 . . . . . . . . . 10  |-  ( ( ( f " x
)  ~~  B  /\  B  ~~  A )  -> 
( f " x
)  ~~  A )
3327, 31, 32syl2anc 661 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  A
)
34 fin4i 8674 . . . . . . . . 9  |-  ( ( ( f " x
)  C.  A  /\  ( f " x
)  ~~  A )  ->  -.  A  e. FinIV )
3520, 33, 34syl2anc 661 . . . . . . . 8  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
3635ex 434 . . . . . . 7  |-  ( f : B -1-1-onto-> A  ->  ( (
x  C.  B  /\  x  ~~  B )  ->  -.  A  e. FinIV ) )
3736exlimdv 1700 . . . . . 6  |-  ( f : B -1-1-onto-> A  ->  ( E. x ( x  C.  B  /\  x  ~~  B
)  ->  -.  A  e. FinIV
) )
3837con2d 115 . . . . 5  |-  ( f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
3938exlimiv 1698 . . . 4  |-  ( E. f  f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B
) ) )
402, 39sylbi 195 . . 3  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  -.  E. x
( x  C.  B  /\  x  ~~  B ) ) )
41 relen 7518 . . . . 5  |-  Rel  ~~
4241brrelexi 5039 . . . 4  |-  ( B 
~~  A  ->  B  e.  _V )
43 isfin4 8673 . . . 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 234 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    C_ wss 3476    C. wpss 3477   class class class wbr 4447   dom cdm 4999   ran crn 5000   "cima 5002   -1-1->wf1 5583   -1-1-onto->wf1o 5585    ~~ cen 7510  FinIVcfin4 8656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-fin4 8663
This theorem is referenced by:  domfin4  8687  isfin4-3  8691
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