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Theorem canth4 8264
Description: An "effective" form of Cantor's theorem canth 6287. 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 2284 . . . . . . . 8  |-  B  =  B
2 eqid 2284 . . . . . . . 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 2797 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  _V )
653ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  A  e.  _V )
7 simpl2 961 . . . . . . . . 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 959 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( ~P A  i^i  dom 
card )  C_  D
)
98sselda 3181 . . . . . . . . 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 5624 . . . . . . . . 9  |-  ( ( F : D --> A  /\  x  e.  D )  ->  ( F `  x
)  e.  A )
117, 9, 10syl2anc 644 . . . . . . . 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 8263 . . . . . . 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 8262 . . . . 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 5032 . . . . 5  |-  ( `' ( W `  B
) " { ( F `  B ) } )  C_  dom  (  W `  B )
2220, 21eqsstri 3209 . . . 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 4877 . . . . . 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 4897 . . . . 5  |-  dom  (  B  X.  B )  =  B
2725, 26syl6sseq 3225 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  (  W `  B )  C_  B
)
2822, 27syl5ss 3191 . . 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 4383 . . . . . . . 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 4334 . . . . . . 7  |-  ( ( ( W `  B
)  Or  B  /\  ( F `  B )  e.  B )  ->  -.  ( F `  B
) ( W `  B ) ( F `
 B ) )
3533, 29, 34syl2anc 644 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B ) ( W `
 B ) ( F `  B ) )
3620eleq2i 2348 . . . . . . 7  |-  ( ( F `  B )  e.  C  <->  ( F `  B )  e.  ( `' ( W `  B ) " {
( F `  B
) } ) )
37 fvex 5499 . . . . . . . 8  |-  ( F `
 B )  e. 
_V
3837eliniseg 5041 . . . . . . . 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 2536 . . . . 5  |-  ( ( ( F `  B
)  e.  B  /\  -.  ( F `  B
)  e.  C )  ->  B  =/=  C
)
4329, 41, 42syl2anc 644 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B  =/=  C )
4443necomd 2530 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  =/=  B )
45 df-pss 3169 . . 3  |-  ( C 
C.  B  <->  ( C  C_  B  /\  C  =/= 
B ) )
4628, 44, 45sylanbrc 647 . 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 3652 . . . . . . . . 9  |-  ( y  =  ( F `  B )  ->  { y }  =  { ( F `  B ) } )
4948imaeq2d 5011 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  ( `' ( W `
 B ) " { ( F `  B ) } ) )
5049, 20syl6eqr 2334 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  C )
5150fveq2d 5489 . . . . . 6  |-  ( y  =  ( F `  B )  ->  ( F `  ( `' ( W `  B )
" { y } ) )  =  ( F `  C ) )
52 id 21 . . . . . 6  |-  ( y  =  ( F `  B )  ->  y  =  ( F `  B ) )
5351, 52eqeq12d 2298 . . . . 5  |-  ( y  =  ( F `  B )  ->  (
( F `  ( `' ( W `  B ) " {
y } ) )  =  y  <->  ( F `  C )  =  ( F `  B ) ) )
5453rspcv 2881 . . . 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 2289 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  =  ( F `
 C ) )
5719, 46, 563jca 1134 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   _Vcvv 2789    i^i cin 3152    C_ wss 3153    C. wpss 3154   ~Pcpw 3626   {csn 3641   U.cuni 3828   class class class wbr 4024   {copab 4077    Or wor 4312    We wwe 4350    X. cxp 4686   `'ccnv 4687   dom cdm 4688   "cima 4691   -->wf 5217   ` cfv 5221   cardccrd 7563
This theorem is referenced by:  canthnumlem  8265  canthp1lem2  8270
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-1st 6083  df-iota 6252  df-riota 6299  df-recs 6383  df-en 6859  df-oi 7220  df-card 7567
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