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Theorem canthnumlem 8487
Description: Lemma for canthnum 8488. (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 5606 . . . . 5  |-  ( F : ( ~P A  i^i  dom  card ) -1-1-> A  ->  F : ( ~P A  i^i  dom  card ) --> A )
2 ssid 3335 . . . . . 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 8486 . . . . . 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 1268 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) --> A )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B
)  =  ( F `
 C ) ) )
81, 7sylan2 461 . . . 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 971 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( F `  B )  =  ( F `  C ) )
10 simpr 448 . . . 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 969 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  C_  A
)
12 elpw2g 4331 . . . . . . 7  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
1312adantr 452 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B  e. 
~P A  <->  B  C_  A
) )
1411, 13mpbird 224 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ~P A )
15 eqid 2412 . . . . . . . . . . . . 13  |-  B  =  B
16 eqid 2412 . . . . . . . . . . . . 13  |-  ( W `
 B )  =  ( W `  B
)
1715, 16pm3.2i 442 . . . . . . . . . . . 12  |-  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) )
18 elex 2932 . . . . . . . . . . . . . 14  |-  ( A  e.  V  ->  A  e.  _V )
1918adantr 452 . . . . . . . . . . . . 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 5837 . . . . . . . . . . . . 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 8485 . . . . . . . . . . . 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 225 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( B W ( W `  B
)  /\  ( F `  B )  e.  B
) )
2423simpld 446 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B W ( W `  B ) )
253, 19fpwwelem 8484 . . . . . . . . . 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 202 . . . . . . . . 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 450 . . . . . . . 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 446 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( W `  B )  We  B
)
29 fvex 5709 . . . . . . . 8  |-  ( W `
 B )  e. 
_V
30 weeq1 4538 . . . . . . . 8  |-  ( r  =  ( W `  B )  ->  (
r  We  B  <->  ( W `  B )  We  B
) )
3129, 30spcev 3011 . . . . . . 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 7880 . . . . . 6  |-  ( B  e.  dom  card  <->  E. r 
r  We  B )
3432, 33sylibr 204 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  dom  card )
35 elin 3498 . . . . 5  |-  ( B  e.  ( ~P A  i^i  dom  card )  <->  ( B  e.  ~P A  /\  B  e.  dom  card ) )
3614, 34, 35sylanbrc 646 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  e.  ( ~P A  i^i  dom  card ) )
378simp2d 970 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C.  B
)
3837pssssd 3412 . . . . . . 7  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  B
)
3938, 11sstrd 3326 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  C_  A
)
40 elpw2g 4331 . . . . . . 7  |-  ( A  e.  V  ->  ( C  e.  ~P A  <->  C 
C_  A ) )
4140adantr 452 . . . . . 6  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( C  e. 
~P A  <->  C  C_  A
) )
4239, 41mpbird 224 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ~P A )
43 ssnum 7884 . . . . . 6  |-  ( ( B  e.  dom  card  /\  C  C_  B )  ->  C  e.  dom  card )
4434, 38, 43syl2anc 643 . . . . 5  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  dom  card )
45 elin 3498 . . . . 5  |-  ( C  e.  ( ~P A  i^i  dom  card )  <->  ( C  e.  ~P A  /\  C  e.  dom  card ) )
4642, 44, 45sylanbrc 646 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  e.  ( ~P A  i^i  dom  card ) )
47 f1fveq 5975 . . . 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 ) )
4810, 36, 46, 47syl12anc 1182 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  ( ( F `
 B )  =  ( F `  C
)  <->  B  =  C
) )
499, 48mpbid 202 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =  C )
5037pssned 3413 . . . 4  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  C  =/=  B
)
5150necomd 2658 . . 3  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  B  =/=  C
)
5251neneqd 2591 . 2  |-  ( ( A  e.  V  /\  F : ( ~P A  i^i  dom  card ) -1-1-> A )  ->  -.  B  =  C )
5349, 52pm2.65da 560 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    i^i cin 3287    C_ wss 3288    C. wpss 3289   ~Pcpw 3767   {csn 3782   U.cuni 3983   class class class wbr 4180   {copab 4233    We wwe 4508    X. cxp 4843   `'ccnv 4844   dom cdm 4845   "cima 4848   -->wf 5417   -1-1->wf1 5418   ` cfv 5421   cardccrd 7786
This theorem is referenced by:  canthnum  8488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-1st 6316  df-riota 6516  df-recs 6600  df-er 6872  df-en 7077  df-dom 7078  df-oi 7443  df-card 7790
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