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Theorem fphpd 35368
Description: Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
fphpd.a  |-  ( ph  ->  B  ~<  A )
fphpd.b  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
fphpd.c  |-  ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
fphpd  |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D
) )
Distinct variable groups:    x, A, y    x, B, y    y, C    x, D    ph, x, y
Allowed substitution hints:    C( x)    D( y)

Proof of Theorem fphpd
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 domnsym 7704 . . . 4  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
2 fphpd.a . . . 4  |-  ( ph  ->  B  ~<  A )
31, 2nsyl3 122 . . 3  |-  ( ph  ->  -.  A  ~<_  B )
4 relsdom 7584 . . . . . . 7  |-  Rel  ~<
54brrelexi 4895 . . . . . 6  |-  ( B 
~<  A  ->  B  e. 
_V )
62, 5syl 17 . . . . 5  |-  ( ph  ->  B  e.  _V )
76adantr 466 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  B  e.  _V )
8 nfv 1754 . . . . . . . . 9  |-  F/ x
( ph  /\  a  e.  A )
9 nfcsb1v 3417 . . . . . . . . . 10  |-  F/_ x [_ a  /  x ]_ C
109nfel1 2607 . . . . . . . . 9  |-  F/ x [_ a  /  x ]_ C  e.  B
118, 10nfim 1978 . . . . . . . 8  |-  F/ x
( ( ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B
)
12 eleq1 2501 . . . . . . . . . 10  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
1312anbi2d 708 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  a  e.  A ) ) )
14 csbeq1a 3410 . . . . . . . . . 10  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1514eleq1d 2498 . . . . . . . . 9  |-  ( x  =  a  ->  ( C  e.  B  <->  [_ a  /  x ]_ C  e.  B
) )
1613, 15imbi12d 321 . . . . . . . 8  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  A )  ->  C  e.  B )  <-> 
( ( ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B
) ) )
17 fphpd.b . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
1811, 16, 17chvar 2069 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B )
1918ex 435 . . . . . 6  |-  ( ph  ->  ( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
2019adantr 466 . . . . 5  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
21 csbid 3409 . . . . . . . . . . 11  |-  [_ x  /  x ]_ C  =  C
22 vex 3090 . . . . . . . . . . . 12  |-  y  e. 
_V
23 fphpd.c . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  D )
2422, 23csbie 3427 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  D
2521, 24eqeq12i 2449 . . . . . . . . . 10  |-  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  C  =  D
)
2625imbi1i 326 . . . . . . . . 9  |-  ( (
[_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  ( C  =  D  ->  x  =  y )
)
27262ralbii 2864 . . . . . . . 8  |-  ( A. x  e.  A  A. y  e.  A  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
28 nfcsb1v 3417 . . . . . . . . . . . 12  |-  F/_ x [_ y  /  x ]_ C
299, 28nfeq 2602 . . . . . . . . . . 11  |-  F/ x [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
30 nfv 1754 . . . . . . . . . . 11  |-  F/ x  a  =  y
3129, 30nfim 1978 . . . . . . . . . 10  |-  F/ x
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )
32 nfv 1754 . . . . . . . . . 10  |-  F/ y ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  ->  a  =  b )
33 csbeq1 3404 . . . . . . . . . . . 12  |-  ( x  =  a  ->  [_ x  /  x ]_ C  = 
[_ a  /  x ]_ C )
3433eqeq1d 2431 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
) )
35 equequ1 1850 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
3634, 35imbi12d 321 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  (
[_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y ) ) )
37 csbeq1 3404 . . . . . . . . . . . 12  |-  ( y  =  b  ->  [_ y  /  x ]_ C  = 
[_ b  /  x ]_ C )
3837eqeq2d 2443 . . . . . . . . . . 11  |-  ( y  =  b  ->  ( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ b  /  x ]_ C
) )
39 equequ2 1851 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
4038, 39imbi12d 321 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )  <-> 
( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4131, 32, 36, 40rspc2 3196 . . . . . . . . 9  |-  ( ( a  e.  A  /\  b  e.  A )  ->  ( A. x  e.  A  A. y  e.  A  ( [_ x  /  x ]_ C  = 
[_ y  /  x ]_ C  ->  x  =  y )  ->  ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4241com12 32 . . . . . . . 8  |-  ( A. x  e.  A  A. y  e.  A  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  ->  ( ( a  e.  A  /\  b  e.  A )  ->  ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4327, 42sylbir 216 . . . . . . 7  |-  ( A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y )  -> 
( ( a  e.  A  /\  b  e.  A )  ->  ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
44 id 23 . . . . . . . 8  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  ->  a  =  b ) )
45 csbeq1 3404 . . . . . . . 8  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
4644, 45impbid1 206 . . . . . . 7  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  <->  a  =  b ) )
4743, 46syl6 34 . . . . . 6  |-  ( A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y )  -> 
( ( a  e.  A  /\  b  e.  A )  ->  ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  <->  a  =  b ) ) )
4847adantl 467 . . . . 5  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( ( a  e.  A  /\  b  e.  A )  ->  ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  <->  a  =  b ) ) )
4920, 48dom2d 7617 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( B  e.  _V  ->  A  ~<_  B ) )
507, 49mpd 15 . . 3  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  A  ~<_  B )
513, 50mtand 663 . 2  |-  ( ph  ->  -.  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
52 ancom 451 . . . . . . 7  |-  ( ( -.  x  =  y  /\  C  =  D )  <->  ( C  =  D  /\  -.  x  =  y ) )
53 df-ne 2627 . . . . . . . 8  |-  ( x  =/=  y  <->  -.  x  =  y )
5453anbi1i 699 . . . . . . 7  |-  ( ( x  =/=  y  /\  C  =  D )  <->  ( -.  x  =  y  /\  C  =  D ) )
55 pm4.61 427 . . . . . . 7  |-  ( -.  ( C  =  D  ->  x  =  y )  <->  ( C  =  D  /\  -.  x  =  y ) )
5652, 54, 553bitr4i 280 . . . . . 6  |-  ( ( x  =/=  y  /\  C  =  D )  <->  -.  ( C  =  D  ->  x  =  y ) )
5756rexbii 2934 . . . . 5  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  E. y  e.  A  -.  ( C  =  D  ->  x  =  y ) )
58 rexnal 2880 . . . . 5  |-  ( E. y  e.  A  -.  ( C  =  D  ->  x  =  y )  <->  -.  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
5957, 58bitri 252 . . . 4  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  -. 
A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6059rexbii 2934 . . 3  |-  ( E. x  e.  A  E. y  e.  A  (
x  =/=  y  /\  C  =  D )  <->  E. x  e.  A  -.  A. y  e.  A  ( C  =  D  ->  x  =  y )
)
61 rexnal 2880 . . 3  |-  ( E. x  e.  A  -.  A. y  e.  A  ( C  =  D  ->  x  =  y )  <->  -. 
A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6260, 61bitri 252 . 2  |-  ( E. x  e.  A  E. y  e.  A  (
x  =/=  y  /\  C  =  D )  <->  -. 
A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6351, 62sylibr 215 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087   [_csb 3401   class class class wbr 4426    ~<_ cdom 7575    ~< csdm 7576
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580
This theorem is referenced by:  fphpdo  35369  pellex  35389
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