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

Theorem canthnumlem 8927
Description: Lemma for canthnum 8928. (Contributed by Mario Carneiro, 19-May-2015.)
Hypotheses
Ref Expression
canth4.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
canth4.2  |-  B  = 
U. dom  W
canth4.3  |-  C  =  ( `' ( W `
 B ) " { ( F `  B ) } )
Assertion
Ref Expression
canthnumlem  |-  ( A  e.  V  ->  -.  F : ( ~P A  i^i  dom  card ) -1-1-> A )
Distinct variable groups:    x, r,
y, A    B, r, x, y    F, r, x, y    V, r, x, y   
y, C    W, r, x, y
Allowed substitution hints:    C( x, r)

Proof of Theorem canthnumlem
StepHypRef Expression
1 f1f 5715 . . . . 5  |-  ( F : ( ~P A  i^i  dom  card ) -1-1-> A  ->  F : ( ~P A  i^i  dom  card ) --> A )
2 ssid 3484 . . . . . 6  |-  ( ~P A  i^i  dom  card )  C_  ( ~P A  i^i  dom  card )
3 canth4.1 . . . . . . 7  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
4 canth4.2 . . . . . . 7  |-  B  = 
U. dom  W
5 canth4.3 . . . . . . 7  |-  C  =  ( `' ( W `
 B ) " { ( F `  B ) } )
63, 4, 5canth4 8926 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) --> A  /\  ( ~P A  i^i  dom  card )  C_  ( ~P A  i^i  dom  card ) )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
72, 6mp3an3 1304 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) --> A )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
81, 7sylan2 474 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
98simp3d 1002 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( F `  B )  =  ( F `  C ) )
10 simpr 461 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  F : ( ~P A  i^i  dom  card ) -1-1-> A )
118simp1d 1000 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  C_  A
)
12 elpw2g 4564 . . . . . . 7  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
1312adantr 465 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B  e. 
~P A  <->  B  C_  A
) )
1411, 13mpbird 232 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ~P A )
15 eqid 2454 . . . . . . . . . . . . 13  |-  B  =  B
16 eqid 2454 . . . . . . . . . . . . 13  |-  ( W `
 B )  =  ( W `  B
)
1715, 16pm3.2i 455 . . . . . . . . . . . 12  |-  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) )
18 elex 3087 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  A  e.  _V )
1918adantr 465 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  A  e.  _V )
2010, 1syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  F : ( ~P A  i^i  dom  card ) --> A )
2120ffvelrnda 5953 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A
)
223, 19, 21, 4fpwwe 8925 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( B W ( W `  B )  /\  ( F `  B )  e.  B )  <->  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) ) ) )
2317, 22mpbiri 233 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B W ( W `  B
)  /\  ( F `  B )  e.  B
) )
2423simpld 459 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B W ( W `  B ) )
253, 19fpwwelem 8924 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B W ( W `  B
)  <->  ( ( B 
C_  A  /\  ( W `  B )  C_  ( B  X.  B
) )  /\  (
( W `  B
)  We  B  /\  A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y ) ) ) )
2624, 25mpbid 210 . . . . . . . . 9  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( B 
C_  A  /\  ( W `  B )  C_  ( B  X.  B
) )  /\  (
( W `  B
)  We  B  /\  A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y ) ) )
2726simprd 463 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( W `
 B )  We  B  /\  A. y  e.  B  ( F `  ( `' ( W `
 B ) " { y } ) )  =  y ) )
2827simpld 459 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( W `  B )  We  B
)
29 fvex 5810 . . . . . . . 8  |-  ( W `
 B )  e. 
_V
30 weeq1 4817 . . . . . . . 8  |-  ( r  =  ( W `  B )  ->  (
r  We  B  <->  ( W `  B )  We  B
) )
3129, 30spcev 3170 . . . . . . 7  |-  ( ( W `  B )  We  B  ->  E. r 
r  We  B )
3228, 31syl 16 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  E. r  r  We  B )
33 ween 8317 . . . . . 6  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
3432, 33sylibr 212 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  dom  card )
3514, 34elind 3649 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ( ~P A  i^i  dom  card ) )
368simp2d 1001 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C.  B
)
3736pssssd 3562 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  B
)
3837, 11sstrd 3475 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  A
)
39 elpw2g 4564 . . . . . . 7  |-  ( A  e.  V  ->  ( C  e.  ~P A  <->  C 
C_  A ) )
4039adantr 465 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( C  e. 
~P A  <->  C  C_  A
) )
4138, 40mpbird 232 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ~P A )
42 ssnum 8321 . . . . . 6  |-  ( ( B  e.  dom  card  /\  C  C_  B )  ->  C  e.  dom  card )
4334, 37, 42syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  dom  card )
4441, 43elind 3649 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ( ~P A  i^i  dom  card ) )
45 f1fveq 6085 . . . 4  |-  ( ( F : ( ~P A  i^i  dom  card ) -1-1-> A  /\  ( B  e.  ( ~P A  i^i  dom  card )  /\  C  e.  ( ~P A  i^i  dom  card ) ) )  ->  ( ( F `  B )  =  ( F `  C )  <->  B  =  C ) )
4610, 35, 44, 45syl12anc 1217 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( F `
 B )  =  ( F `  C
)  <->  B  =  C
) )
479, 46mpbid 210 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =  C )
4836pssned 3563 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  =/=  B
)
4948necomd 2723 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =/=  C
)
5049neneqd 2655 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  -.  B  =  C )
5147, 50pm2.65da 576 1  |-  ( A  e.  V  ->  -.  F : ( ~P A  i^i  dom  card ) -1-1-> A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2799   _Vcvv 3078    i^i cin 3436    C_ wss 3437    C. wpss 3438   ~Pcpw 3969   {csn 3986   U.cuni 4200   class class class wbr 4401   {copab 4458    We wwe 4787    X. cxp 4947   `'ccnv 4948   dom cdm 4949   "cima 4952   -->wf 5523   -1-1->wf1 5524   ` cfv 5527   cardccrd 8217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-1st 6688  df-recs 6943  df-er 7212  df-en 7422  df-dom 7423  df-oi 7836  df-card 8221
This theorem is referenced by:  canthnum  8928
  Copyright terms: Public domain W3C validator