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Theorem canthnumlem 8272
Description: Lemma for canthnum 8273. (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 5439 . . . . 5  |-  ( F : ( ~P A  i^i  dom  card ) -1-1-> A  ->  F : ( ~P A  i^i  dom  card ) --> A )
2 ssid 3199 . . . . . 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 8271 . . . . . 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 1266 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) --> A )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
81, 7sylan2 460 . . . 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 969 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( F `  B )  =  ( F `  C ) )
10 simpr 447 . . . 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 967 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  C_  A
)
12 elpw2g 4176 . . . . . . 7  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
1312adantr 451 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B  e. 
~P A  <->  B  C_  A
) )
1411, 13mpbird 223 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ~P A )
15 eqid 2285 . . . . . . . . . . . . 13  |-  B  =  B
16 eqid 2285 . . . . . . . . . . . . 13  |-  ( W `
 B )  =  ( W `  B
)
1715, 16pm3.2i 441 . . . . . . . . . . . 12  |-  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) )
18 elex 2798 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  A  e.  _V )
1918adantr 451 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  A  e.  _V )
2010, 1syl 15 . . . . . . . . . . . . . 14  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  F : ( ~P A  i^i  dom  card ) --> A )
21 ffvelrn 5665 . . . . . . . . . . . . . 14  |-  ( ( F : ( ~P A  i^i  dom  card )
--> A  /\  x  e.  ( ~P A  i^i  dom 
card ) )  -> 
( F `  x
)  e.  A )
2220, 21sylan 457 . . . . . . . . . . . . 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
)
233, 19, 22, 4fpwwe 8270 . . . . . . . . . . . 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 ) ) ) )
2417, 23mpbiri 224 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B W ( W `  B
)  /\  ( F `  B )  e.  B
) )
2524simpld 445 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B W ( W `  B ) )
263, 19fpwwelem 8269 . . . . . . . . . 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 ) ) ) )
2725, 26mpbid 201 . . . . . . . . 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 ) ) )
2827simprd 449 . . . . . . . 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 ) )
2928simpld 445 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( W `  B )  We  B
)
30 fvex 5541 . . . . . . . 8  |-  ( W `
 B )  e. 
_V
31 weeq1 4383 . . . . . . . 8  |-  ( r  =  ( W `  B )  ->  (
r  We  B  <->  ( W `  B )  We  B
) )
3230, 31spcev 2877 . . . . . . 7  |-  ( ( W `  B )  We  B  ->  E. r 
r  We  B )
3329, 32syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  E. r  r  We  B )
34 ween 7664 . . . . . 6  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
3533, 34sylibr 203 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  dom  card )
36 elin 3360 . . . . 5  |-  ( B  e.  ( ~P A  i^i  dom  card )  <->  ( B  e.  ~P A  /\  B  e.  dom  card ) )
3714, 35, 36sylanbrc 645 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ( ~P A  i^i  dom  card ) )
388simp2d 968 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C.  B
)
3938pssssd 3275 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  B
)
4039, 11sstrd 3191 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  A
)
41 elpw2g 4176 . . . . . . 7  |-  ( A  e.  V  ->  ( C  e.  ~P A  <->  C 
C_  A ) )
4241adantr 451 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( C  e. 
~P A  <->  C  C_  A
) )
4340, 42mpbird 223 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ~P A )
44 ssnum 7668 . . . . . 6  |-  ( ( B  e.  dom  card  /\  C  C_  B )  ->  C  e.  dom  card )
4535, 39, 44syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  dom  card )
46 elin 3360 . . . . 5  |-  ( C  e.  ( ~P A  i^i  dom  card )  <->  ( C  e.  ~P A  /\  C  e.  dom  card ) )
4743, 45, 46sylanbrc 645 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ( ~P A  i^i  dom  card ) )
48 f1fveq 5788 . . . 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 ) )
4910, 37, 47, 48syl12anc 1180 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( F `
 B )  =  ( F `  C
)  <->  B  =  C
) )
509, 49mpbid 201 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =  C )
51 pssne 3274 . . . . 5  |-  ( C 
C.  B  ->  C  =/=  B )
5238, 51syl 15 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  =/=  B
)
5352necomd 2531 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =/=  C
)
5453neneqd 2464 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  -.  B  =  C )
5550, 54pm2.65da 559 1  |-  ( A  e.  V  ->  -.  F : ( ~P A  i^i  dom  card ) -1-1-> A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   _Vcvv 2790    i^i cin 3153    C_ wss 3154    C. wpss 3155   ~Pcpw 3627   {csn 3642   U.cuni 3829   class class class wbr 4025   {copab 4078    We wwe 4353    X. cxp 4689   `'ccnv 4690   dom cdm 4691   "cima 4694   -->wf 5253   -1-1->wf1 5254   ` cfv 5257   cardccrd 7570
This theorem is referenced by:  canthnum  8273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-1st 6124  df-riota 6306  df-recs 6390  df-er 6662  df-en 6866  df-dom 6867  df-oi 7227  df-card 7574
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