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Theorem canth4 8514
Description: An "effective" form of Cantor's theorem canth 6531. 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 2435 . . . . . . . 8  |-  B  =  B
2 eqid 2435 . . . . . . . 8  |-  ( W `
 B )  =  ( W `  B
)
31, 2pm3.2i 442 . . . . . . 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 2956 . . . . . . . . 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 3340 . . . . . . . . 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 5863 . . . . . . . 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 8513 . . . . . . 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 225 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B W ( W `  B )  /\  ( F `  B )  e.  B
) )
1413simpld 446 . . . . 5  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  B W ( W `
 B ) )
154, 6fpwwelem 8512 . . . . 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 202 . . . 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 446 . . 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 446 . 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 5216 . . . . 5  |-  ( `' ( W `  B
) " { ( F `  B ) } )  C_  dom  ( W `  B )
2119, 20eqsstri 3370 . . . 4  |-  C  C_  dom  ( W `  B
)
2217simprd 450 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  C_  ( B  X.  B ) )
23 dmss 5061 . . . . . 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 5081 . . . . 5  |-  dom  ( B  X.  B )  =  B
2624, 25syl6sseq 3386 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  dom  ( W `  B )  C_  B
)
2721, 26syl5ss 3351 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C_  B )
2813simprd 450 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  e.  B )
2916simprd 450 . . . . . . 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 446 . . . . . 6  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( W `  B
)  We  B )
31 weso 4565 . . . . . 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 4516 . . . . 5  |-  ( ( ( W `  B
)  Or  B  /\  ( F `  B )  e.  B )  ->  -.  ( F `  B
) ( W `  B ) ( F `
 B ) )
3432, 28, 33syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B ) ( W `
 B ) ( F `  B ) )
3519eleq2i 2499 . . . . 5  |-  ( ( F `  B )  e.  C  <->  ( F `  B )  e.  ( `' ( W `  B ) " {
( F `  B
) } ) )
36 fvex 5734 . . . . . 6  |-  ( F `
 B )  e. 
_V
3736eliniseg 5225 . . . . . 6  |-  ( ( F `  B )  e.  _V  ->  (
( F `  B
)  e.  ( `' ( W `  B
) " { ( F `  B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) ) )
3836, 37ax-mp 8 . . . . 5  |-  ( ( F `  B )  e.  ( `' ( W `  B )
" { ( F `
 B ) } )  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
3935, 38bitri 241 . . . 4  |-  ( ( F `  B )  e.  C  <->  ( F `  B ) ( W `
 B ) ( F `  B ) )
4034, 39sylnibr 297 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  -.  ( F `  B )  e.  C
)
4127, 28, 40ssnelpssd 3684 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  C  C.  B )
4229simprd 450 . . . 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 3817 . . . . . . . . 9  |-  ( y  =  ( F `  B )  ->  { y }  =  { ( F `  B ) } )
4443imaeq2d 5195 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  ( `' ( W `
 B ) " { ( F `  B ) } ) )
4544, 19syl6eqr 2485 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  ( `' ( W `  B ) " {
y } )  =  C )
4645fveq2d 5724 . . . . . 6  |-  ( y  =  ( F `  B )  ->  ( F `  ( `' ( W `  B )
" { y } ) )  =  ( F `  C ) )
47 id 20 . . . . . 6  |-  ( y  =  ( F `  B )  ->  y  =  ( F `  B ) )
4846, 47eqeq12d 2449 . . . . 5  |-  ( y  =  ( F `  B )  ->  (
( F `  ( `' ( W `  B ) " {
y } ) )  =  y  <->  ( F `  C )  =  ( F `  B ) ) )
4948rspcv 3040 . . . 4  |-  ( ( F `  B )  e.  B  ->  ( A. y  e.  B  ( F `  ( `' ( W `  B
) " { y } ) )  =  y  ->  ( F `  C )  =  ( F `  B ) ) )
5028, 42, 49sylc 58 . . 3  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  C
)  =  ( F `
 B ) )
5150eqcomd 2440 . 2  |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( F `  B
)  =  ( F `
 C ) )
5218, 41, 513jca 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312    C. wpss 3313   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204   {copab 4257    Or wor 4494    We wwe 4532    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873   -->wf 5442   ` cfv 5446   cardccrd 7814
This theorem is referenced by:  canthnumlem  8515  canthp1lem2  8520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-1st 6341  df-riota 6541  df-recs 6625  df-en 7102  df-oi 7471  df-card 7818
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