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Theorem canth4 8223
Description: An "effective" form of Cantor's theorem canth 6246. For any function  F from the powerset of  A to  A, there are two definable sets  B and  C which witness non-injectivity of  F. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-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
canth4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B )  =  ( F `  C ) ) )
Distinct variable groups:    x, r,
y, A    B, r, x, y    D, 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 canth4
StepHypRef Expression
1 eqid 2256 . . . . . . . 8  |-  B  =  B
2 eqid 2256 . . . . . . . 8  |-  ( W `
 B )  =  ( W `  B
)
31, 2pm3.2i 443 . . . . . . 7  |-  ( B  =  B  /\  ( W `  B )  =  ( W `  B ) )
4 canth4.1 . . . . . . . 8  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
5 elex 2765 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
653ad2ant1 981 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A  e.  _V )
7 simpl2 964 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  F : D --> A )
8 simp3 962 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ~P A  i^i  dom 
card )  C_  D
)
98sselda 3141 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  x  e.  D )
10 ffvelrn 5583 . . . . . . . . 9  |-  ( ( F : D --> A  /\  x  e.  D )  ->  ( F `  x
)  e.  A )
117, 9, 10syl2anc 645 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )
12 canth4.2 . . . . . . . 8  |-  B  = 
U. dom  W
134, 6, 11, 12fpwwe 8222 . . . . . . 7  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ( B W ( W `  B
)  /\  ( F `  B )  e.  B
)  <->  ( B  =  B  /\  ( W `
 B )  =  ( W `  B
) ) ) )
143, 13mpbiri 226 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B W ( W `  B )  /\  ( F `  B )  e.  B
) )
1514simpld 447 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B W ( W `
 B ) )
164, 6fpwwelem 8221 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( 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 ) ) ) )
1715, 16mpbid 203 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ( B  C_  A  /\  ( W `  B )  C_  ( B  X.  B ) )  /\  ( ( W `
 B )  We  B  /\  A. y  e.  B  ( F `  ( `' ( W `
 B ) " { y } ) )  =  y ) ) )
1817simpld 447 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  ( W `  B
)  C_  ( B  X.  B ) ) )
1918simpld 447 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B  C_  A )
20 canth4.3 . . . . 5  |-  C  =  ( `' ( W `
 B ) " { ( F `  B ) } )
21 cnvimass 5007 . . . . 5  |-  ( `' ( W `  B
) " { ( F `  B ) } )  C_  dom  (  W `  B )
2220, 21eqsstri 3169 . . . 4  |-  C  C_  dom  (  W `  B
)
2318simprd 451 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  C_  ( B  X.  B ) )
24 dmss 4852 . . . . . 6  |-  ( ( W `  B ) 
C_  ( B  X.  B )  ->  dom  (  W `  B ) 
C_  dom  (  B  X.  B ) )
2523, 24syl 17 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  (  W `  B )  C_  dom  (  B  X.  B
) )
26 dmxpid 4872 . . . . 5  |-  dom  (  B  X.  B )  =  B
2725, 26syl6sseq 3185 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  (  W `  B )  C_  B
)
2822, 27syl5ss 3151 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C_  B )
2914simprd 451 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  e.  B )
3017simprd 451 . . . . . . . . 9  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ( W `  B )  We  B  /\  A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y ) )
3130simpld 447 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  We  B )
32 weso 4342 . . . . . . . 8  |-  ( ( W `  B )  We  B  ->  ( W `  B )  Or  B )
3331, 32syl 17 . . . . . . 7  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  Or  B )
34 sonr 4293 . . . . . . 7  |-  ( ( ( W `  B
)  Or  B  /\  ( F `  B )  e.  B )  ->  -.  ( F `  B
) ( W `  B ) ( F `
 B ) )
3533, 29, 34syl2anc 645 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B ) ( W `
 B ) ( F `  B ) )
3620eleq2i 2320 . . . . . . 7  |-  ( ( F `  B )  e.  C  <->  ( F `  B )  e.  ( `' ( W `  B ) " {
( F `  B
) } ) )
37 fvex 5458 . . . . . . . 8  |-  ( F `
 B )  e. 
_V
3837eliniseg 5016 . . . . . . . 8  |-  ( ( F `  B )  e.  _V  ->  (
( F `  B
)  e.  ( `' ( W `  B
) " { ( F `  B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) ) )
3937, 38ax-mp 10 . . . . . . 7  |-  ( ( F `  B )  e.  ( `' ( W `  B )
" { ( F `
 B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
4036, 39bitri 242 . . . . . 6  |-  ( ( F `  B )  e.  C  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
4135, 40sylnibr 298 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B )  e.  C
)
42 nelne1 2508 . . . . 5  |-  ( ( ( F `  B
)  e.  B  /\  -.  ( F `  B
)  e.  C )  ->  B  =/=  C
)
4329, 41, 42syl2anc 645 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B  =/=  C )
4443necomd 2502 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  =/=  B )
45 df-pss 3129 . . 3  |-  ( C 
C.  B  <->  ( C  C_  B  /\  C  =/= 
B ) )
4628, 44, 45sylanbrc 648 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C.  B )
4730simprd 451 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y )
48 sneq 3611 . . . . . . . . 9  |-  ( y  =  ( F `  B )  ->  { y }  =  { ( F `  B ) } )
4948imaeq2d 4986 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  ( `' ( W `
 B ) " { ( F `  B ) } ) )
5049, 20syl6eqr 2306 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  C )
5150fveq2d 5448 . . . . . 6  |-  ( y  =  ( F `  B )  ->  ( F `  ( `' ( W `  B )
" { y } ) )  =  ( F `  C ) )
52 id 21 . . . . . 6  |-  ( y  =  ( F `  B )  ->  y  =  ( F `  B ) )
5351, 52eqeq12d 2270 . . . . 5  |-  ( y  =  ( F `  B )  ->  (
( F `  ( `' ( W `  B ) " {
y } ) )  =  y  <->  ( F `  C )  =  ( F `  B ) ) )
5453rcla4v 2848 . . . 4  |-  ( ( F `  B )  e.  B  ->  ( A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y  ->  ( F `  C )  =  ( F `  B ) ) )
5529, 47, 54sylc 58 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  C
)  =  ( F `
 B ) )
5655eqcomd 2261 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  =  ( F `
 C ) )
5719, 46, 563jca 1137 1  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B )  =  ( F `  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   _Vcvv 2757    i^i cin 3112    C_ wss 3113    C. wpss 3114   ~Pcpw 3585   {csn 3600   U.cuni 3787   class class class wbr 3983   {copab 4036    Or wor 4271    We wwe 4309    X. cxp 4645   `'ccnv 4646   dom cdm 4647   "cima 4650   -->wf 4655   ` cfv 4659   cardccrd 7522
This theorem is referenced by:  canthnumlem  8224  canthp1lem2  8229
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-1st 6042  df-iota 6211  df-riota 6258  df-recs 6342  df-en 6818  df-oi 7179  df-card 7526
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