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Theorem canth4 8814
Description: An "effective" form of Cantor's theorem canth 6049. 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 2443 . . . . . . . 8  |-  B  =  B
2 eqid 2443 . . . . . . . 8  |-  ( W `
 B )  =  ( W `  B
)
31, 2pm3.2i 455 . . . . . . 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 2981 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
653ad2ant1 1009 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A  e.  _V )
7 simpl2 992 . . . . . . . . 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 990 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ~P A  i^i  dom 
card )  C_  D
)
98sselda 3356 . . . . . . . . 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 )
107, 9ffvelrnd 5844 . . . . . . . 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 )
11 canth4.2 . . . . . . . 8  |-  B  = 
U. dom  W
124, 6, 10, 11fpwwe 8813 . . . . . . 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
) ) ) )
133, 12mpbiri 233 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B W ( W `  B )  /\  ( F `  B )  e.  B
) )
1413simpld 459 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B W ( W `
 B ) )
154, 6fpwwelem 8812 . . . . 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 ) ) ) )
1614, 15mpbid 210 . . . 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 ) ) )
1716simpld 459 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  ( W `  B
)  C_  ( B  X.  B ) ) )
1817simpld 459 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B  C_  A )
19 canth4.3 . . . . 5  |-  C  =  ( `' ( W `
 B ) " { ( F `  B ) } )
20 cnvimass 5189 . . . . 5  |-  ( `' ( W `  B
) " { ( F `  B ) } )  C_  dom  ( W `  B )
2119, 20eqsstri 3386 . . . 4  |-  C  C_  dom  ( W `  B
)
2217simprd 463 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  C_  ( B  X.  B ) )
23 dmss 5039 . . . . . 6  |-  ( ( W `  B ) 
C_  ( B  X.  B )  ->  dom  ( W `  B ) 
C_  dom  ( B  X.  B ) )
2422, 23syl 16 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  ( W `  B )  C_  dom  ( B  X.  B
) )
25 dmxpid 5059 . . . . 5  |-  dom  ( B  X.  B )  =  B
2624, 25syl6sseq 3402 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  ( W `  B )  C_  B
)
2721, 26syl5ss 3367 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C_  B )
2813simprd 463 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  e.  B )
2916simprd 463 . . . . . . 7  |-  ( ( 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 ) )
3029simpld 459 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  We  B )
31 weso 4711 . . . . . 6  |-  ( ( W `  B )  We  B  ->  ( W `  B )  Or  B )
3230, 31syl 16 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  Or  B )
33 sonr 4662 . . . . 5  |-  ( ( ( W `  B
)  Or  B  /\  ( F `  B )  e.  B )  ->  -.  ( F `  B
) ( W `  B ) ( F `
 B ) )
3432, 28, 33syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B ) ( W `
 B ) ( F `  B ) )
3519eleq2i 2507 . . . . 5  |-  ( ( F `  B )  e.  C  <->  ( F `  B )  e.  ( `' ( W `  B ) " {
( F `  B
) } ) )
36 fvex 5701 . . . . . 6  |-  ( F `
 B )  e. 
_V
3736eliniseg 5198 . . . . . 6  |-  ( ( F `  B )  e.  _V  ->  (
( F `  B
)  e.  ( `' ( W `  B
) " { ( F `  B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) ) )
3836, 37ax-mp 5 . . . . 5  |-  ( ( F `  B )  e.  ( `' ( W `  B )
" { ( F `
 B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
3935, 38bitri 249 . . . 4  |-  ( ( F `  B )  e.  C  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
4034, 39sylnibr 305 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B )  e.  C
)
4127, 28, 40ssnelpssd 3743 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C.  B )
4229simprd 463 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y )
43 sneq 3887 . . . . . . . . 9  |-  ( y  =  ( F `  B )  ->  { y }  =  { ( F `  B ) } )
4443imaeq2d 5169 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  ( `' ( W `
 B ) " { ( F `  B ) } ) )
4544, 19syl6eqr 2493 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  C )
4645fveq2d 5695 . . . . . 6  |-  ( y  =  ( F `  B )  ->  ( F `  ( `' ( W `  B )
" { y } ) )  =  ( F `  C ) )
47 id 22 . . . . . 6  |-  ( y  =  ( F `  B )  ->  y  =  ( F `  B ) )
4846, 47eqeq12d 2457 . . . . 5  |-  ( y  =  ( F `  B )  ->  (
( F `  ( `' ( W `  B ) " {
y } ) )  =  y  <->  ( F `  C )  =  ( F `  B ) ) )
4948rspcv 3069 . . . 4  |-  ( ( F `  B )  e.  B  ->  ( A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y  ->  ( F `  C )  =  ( F `  B ) ) )
5028, 42, 49sylc 60 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  C
)  =  ( F `
 B ) )
5150eqcomd 2448 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  =  ( F `
 C ) )
5218, 41, 513jca 1168 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    i^i cin 3327    C_ wss 3328    C. wpss 3329   ~Pcpw 3860   {csn 3877   U.cuni 4091   class class class wbr 4292   {copab 4349    Or wor 4640    We wwe 4678    X. cxp 4838   `'ccnv 4839   dom cdm 4840   "cima 4843   -->wf 5414   ` cfv 5418   cardccrd 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-1st 6577  df-recs 6832  df-en 7311  df-oi 7724  df-card 8109
This theorem is referenced by:  canthnumlem  8815  canthp1lem2  8820
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