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

Theorem canthnumlem 9070
Description: Lemma for canthnum 9071. (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 5777 . . . . 5  |-  ( F : ( ~P A  i^i  dom  card ) -1-1-> A  ->  F : ( ~P A  i^i  dom  card ) --> A )
2 ssid 3450 . . . . . 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 9069 . . . . . 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 1352 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) --> A )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
81, 7sylan2 477 . . . 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 1021 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( F `  B )  =  ( F `  C ) )
10 simpr 463 . . . 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 1019 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  C_  A
)
12 elpw2g 4565 . . . . . . 7  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
1312adantr 467 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B  e. 
~P A  <->  B  C_  A
) )
1411, 13mpbird 236 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ~P A )
15 eqid 2450 . . . . . . . . . . . . 13  |-  B  =  B
16 eqid 2450 . . . . . . . . . . . . 13  |-  ( W `
 B )  =  ( W `  B
)
1715, 16pm3.2i 457 . . . . . . . . . . . 12  |-  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) )
18 elex 3053 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  A  e.  _V )
1918adantr 467 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  A  e.  _V )
2010, 1syl 17 . . . . . . . . . . . . . 14  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  F : ( ~P A  i^i  dom  card ) --> A )
2120ffvelrnda 6020 . . . . . . . . . . . . 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 9068 . . . . . . . . . . . 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 237 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B W ( W `  B
)  /\  ( F `  B )  e.  B
) )
2423simpld 461 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B W ( W `  B ) )
253, 19fpwwelem 9067 . . . . . . . . . 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 214 . . . . . . . . 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 465 . . . . . . . 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 461 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( W `  B )  We  B
)
29 fvex 5873 . . . . . . . 8  |-  ( W `
 B )  e. 
_V
30 weeq1 4821 . . . . . . . 8  |-  ( r  =  ( W `  B )  ->  (
r  We  B  <->  ( W `  B )  We  B
) )
3129, 30spcev 3140 . . . . . . 7  |-  ( ( W `  B )  We  B  ->  E. r 
r  We  B )
3228, 31syl 17 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  E. r  r  We  B )
33 ween 8463 . . . . . 6  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
3432, 33sylibr 216 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  dom  card )
3514, 34elind 3617 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ( ~P A  i^i  dom  card ) )
368simp2d 1020 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C.  B
)
3736pssssd 3529 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  B
)
3837, 11sstrd 3441 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  A
)
39 elpw2g 4565 . . . . . . 7  |-  ( A  e.  V  ->  ( C  e.  ~P A  <->  C 
C_  A ) )
4039adantr 467 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( C  e. 
~P A  <->  C  C_  A
) )
4138, 40mpbird 236 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ~P A )
42 ssnum 8467 . . . . . 6  |-  ( ( B  e.  dom  card  /\  C  C_  B )  ->  C  e.  dom  card )
4334, 37, 42syl2anc 666 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  dom  card )
4441, 43elind 3617 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ( ~P A  i^i  dom  card ) )
45 f1fveq 6161 . . . 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 1265 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( F `
 B )  =  ( F `  C
)  <->  B  =  C
) )
479, 46mpbid 214 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =  C )
4836pssned 3530 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  =/=  B
)
4948necomd 2678 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =/=  C
)
5049neneqd 2628 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  -.  B  =  C )
5147, 50pm2.65da 579 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 188    /\ wa 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   A.wral 2736   _Vcvv 3044    i^i cin 3402    C_ wss 3403    C. wpss 3404   ~Pcpw 3950   {csn 3967   U.cuni 4197   class class class wbr 4401   {copab 4459    We wwe 4791    X. cxp 4831   `'ccnv 4832   dom cdm 4833   "cima 4836   -->wf 5577   -1-1->wf1 5578   ` cfv 5581   cardccrd 8366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-1st 6790  df-wrecs 7025  df-recs 7087  df-er 7360  df-en 7567  df-dom 7568  df-oi 8022  df-card 8370
This theorem is referenced by:  canthnum  9071
  Copyright terms: Public domain W3C validator