MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin4en1 Structured version   Unicode version

Theorem fin4en1 8720
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 7601 . 2  |-  ( A 
~~  B  ->  B  ~~  A )
2 bren 7562 . . . 4  |-  ( B 
~~  A  <->  E. f 
f : B -1-1-onto-> A )
3 simpr 459 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  ->  x  C.  B )
4 f1of1 5797 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  f : B -1-1-> A )
5 pssss 3537 . . . . . . . . . . . . . 14  |-  ( x 
C.  B  ->  x  C_  B )
6 ssid 3460 . . . . . . . . . . . . . 14  |-  B  C_  B
75, 6jctir 536 . . . . . . . . . . . . 13  |-  ( x 
C.  B  ->  (
x  C_  B  /\  B  C_  B ) )
8 f1imapss 6154 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  ( x  C_  B  /\  B  C_  B ) )  ->  ( (
f " x ) 
C.  ( f " B )  <->  x  C.  B
) )
94, 7, 8syl2an 475 . . . . . . . . . . . 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 5166 . . . . . . . . . . . . . 14  |-  ( f
" dom  f )  =  ran  f
12 f1odm 5802 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  ->  dom  f  =  B )
1312imaeq2d 5156 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ( f " dom  f )  =  ( f " B
) )
14 dff1o5 5807 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  <->  ( f : B -1-1-> A  /\  ran  f  =  A ) )
1514simprbi 462 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ran  f  =  A )
1611, 13, 153eqtr3a 2467 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  ( f " B )  =  A )
1716adantr 463 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " B
)  =  A )
1817psseq2d 3535 . . . . . . . . . . 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 715 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  C.  A
)
21 vex 3061 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221f1imaen 7614 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  x  C_  B )  ->  ( f "
x )  ~~  x
)
234, 5, 22syl2an 475 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  ~~  x )
2423adantrr 715 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  x
)
25 simprr 758 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
26 entr 7604 . . . . . . . . . . 11  |-  ( ( ( f " x
)  ~~  x  /\  x  ~~  B )  -> 
( f " x
)  ~~  B )
2724, 25, 26syl2anc 659 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  B
)
28 vex 3061 . . . . . . . . . . . 12  |-  f  e. 
_V
29 f1oen3g 7568 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : B -1-1-onto-> A )  ->  B  ~~  A )
3028, 29mpan 668 . . . . . . . . . . 11  |-  ( f : B -1-1-onto-> A  ->  B  ~~  A )
3130adantr 463 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  B  ~~  A
)
32 entr 7604 . . . . . . . . . 10  |-  ( ( ( f " x
)  ~~  B  /\  B  ~~  A )  -> 
( f " x
)  ~~  A )
3327, 31, 32syl2anc 659 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  A
)
34 fin4i 8709 . . . . . . . . 9  |-  ( ( ( f " x
)  C.  A  /\  ( f " x
)  ~~  A )  ->  -.  A  e. FinIV )
3520, 33, 34syl2anc 659 . . . . . . . 8  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
3635ex 432 . . . . . . 7  |-  ( f : B -1-1-onto-> A  ->  ( (
x  C.  B  /\  x  ~~  B )  ->  -.  A  e. FinIV ) )
3736exlimdv 1745 . . . . . 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 1743 . . . 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 7558 . . . . 5  |-  Rel  ~~
4241brrelexi 4863 . . . 4  |-  ( B 
~~  A  ->  B  e.  _V )
43 isfin4 8708 . . . 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 234 . 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 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    C_ wss 3413    C. wpss 3414   class class class wbr 4394   dom cdm 4822   ran crn 4823   "cima 4825   -1-1->wf1 5565   -1-1-onto->wf1o 5567    ~~ cen 7550  FinIVcfin4 8691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-er 7347  df-en 7554  df-fin4 8698
This theorem is referenced by:  domfin4  8722  isfin4-3  8726
  Copyright terms: Public domain W3C validator