MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canth4 Structured version   Unicode version

Theorem canth4 9072
Description: An "effective" form of Cantor's theorem canth 6260. 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 2422 . . . . . . . 8  |-  B  =  B
2 eqid 2422 . . . . . . . 8  |-  ( W `
 B )  =  ( W `  B
)
31, 2pm3.2i 456 . . . . . . 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 3090 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
653ad2ant1 1026 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A  e.  _V )
7 simpl2 1009 . . . . . . . . 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 1007 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ~P A  i^i  dom 
card )  C_  D
)
98sselda 3464 . . . . . . . . 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 6034 . . . . . . . 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 9071 . . . . . . 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 236 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B W ( W `  B )  /\  ( F `  B )  e.  B
) )
1413simpld 460 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B W ( W `
 B ) )
154, 6fpwwelem 9070 . . . . 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 213 . . . 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 460 . . 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 460 . 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 5203 . . . . 5  |-  ( `' ( W `  B
) " { ( F `  B ) } )  C_  dom  ( W `  B )
2119, 20eqsstri 3494 . . . 4  |-  C  C_  dom  ( W `  B
)
2217simprd 464 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  C_  ( B  X.  B ) )
23 dmss 5049 . . . . . 6  |-  ( ( W `  B ) 
C_  ( B  X.  B )  ->  dom  ( W `  B ) 
C_  dom  ( B  X.  B ) )
2422, 23syl 17 . . . . 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 5069 . . . . 5  |-  dom  ( B  X.  B )  =  B
2624, 25syl6sseq 3510 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  ( W `  B )  C_  B
)
2721, 26syl5ss 3475 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C_  B )
2813simprd 464 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  e.  B )
2916simprd 464 . . . . . . 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 460 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  We  B )
31 weso 4840 . . . . . 6  |-  ( ( W `  B )  We  B  ->  ( W `  B )  Or  B )
3230, 31syl 17 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  Or  B )
33 sonr 4791 . . . . 5  |-  ( ( ( W `  B
)  Or  B  /\  ( F `  B )  e.  B )  ->  -.  ( F `  B
) ( W `  B ) ( F `
 B ) )
3432, 28, 33syl2anc 665 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B ) ( W `
 B ) ( F `  B ) )
3519eleq2i 2500 . . . . 5  |-  ( ( F `  B )  e.  C  <->  ( F `  B )  e.  ( `' ( W `  B ) " {
( F `  B
) } ) )
36 fvex 5887 . . . . . 6  |-  ( F `
 B )  e. 
_V
3736eliniseg 5212 . . . . . 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 252 . . . 4  |-  ( ( F `  B )  e.  C  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
4034, 39sylnibr 306 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B )  e.  C
)
4127, 28, 40ssnelpssd 3858 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C.  B )
4229simprd 464 . . . 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 4006 . . . . . . . . 9  |-  ( y  =  ( F `  B )  ->  { y }  =  { ( F `  B ) } )
4443imaeq2d 5183 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  ( `' ( W `
 B ) " { ( F `  B ) } ) )
4544, 19syl6eqr 2481 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  C )
4645fveq2d 5881 . . . . . 6  |-  ( y  =  ( F `  B )  ->  ( F `  ( `' ( W `  B )
" { y } ) )  =  ( F `  C ) )
47 id 23 . . . . . 6  |-  ( y  =  ( F `  B )  ->  y  =  ( F `  B ) )
4846, 47eqeq12d 2444 . . . . 5  |-  ( y  =  ( F `  B )  ->  (
( F `  ( `' ( W `  B ) " {
y } ) )  =  y  <->  ( F `  C )  =  ( F `  B ) ) )
4948rspcv 3178 . . . 4  |-  ( ( F `  B )  e.  B  ->  ( A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y  ->  ( F `  C )  =  ( F `  B ) ) )
5028, 42, 49sylc 62 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  C
)  =  ( F `
 B ) )
5150eqcomd 2430 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  =  ( F `
 C ) )
5218, 41, 513jca 1185 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081    i^i cin 3435    C_ wss 3436    C. wpss 3437   ~Pcpw 3979   {csn 3996   U.cuni 4216   class class class wbr 4420   {copab 4478    Or wor 4769    We wwe 4807    X. cxp 4847   `'ccnv 4848   dom cdm 4849   "cima 4852   -->wf 5593   ` cfv 5597   cardccrd 8370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-1st 6803  df-wrecs 7032  df-recs 7094  df-en 7574  df-oi 8027  df-card 8374
This theorem is referenced by:  canthnumlem  9073  canthp1lem2  9078
  Copyright terms: Public domain W3C validator