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Theorem fphpd 27002
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 7003 . . . 4  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
2 fphpd.a . . . 4  |-  ( ph  ->  B  ~<  A )
31, 2nsyl3 111 . . 3  |-  ( ph  ->  -.  A  ~<_  B )
4 relsdom 6886 . . . . . . 7  |-  Rel  ~<
54brrelexi 4745 . . . . . 6  |-  ( B 
~<  A  ->  B  e. 
_V )
62, 5syl 15 . . . . 5  |-  ( ph  ->  B  e.  _V )
76adantr 451 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  B  e.  _V )
8 nfv 1609 . . . . . . . . 9  |-  F/ x
( ph  /\  a  e.  A )
9 nfcsb1v 3126 . . . . . . . . . 10  |-  F/_ x [_ a  /  x ]_ C
109nfel1 2442 . . . . . . . . 9  |-  F/ x [_ a  /  x ]_ C  e.  B
118, 10nfim 1781 . . . . . . . 8  |-  F/ x
( ( ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B
)
12 eleq1 2356 . . . . . . . . . 10  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
1312anbi2d 684 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  a  e.  A ) ) )
14 csbeq1a 3102 . . . . . . . . . 10  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1514eleq1d 2362 . . . . . . . . 9  |-  ( x  =  a  ->  ( C  e.  B  <->  [_ a  /  x ]_ C  e.  B
) )
1613, 15imbi12d 311 . . . . . . . 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 1939 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B )
1918ex 423 . . . . . 6  |-  ( ph  ->  ( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
2019adantr 451 . . . . 5  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
21 csbid 3101 . . . . . . . . . . 11  |-  [_ x  /  x ]_ C  =  C
22 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
23 nfcv 2432 . . . . . . . . . . . 12  |-  F/_ x D
24 fphpd.c . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  D )
2522, 23, 24csbief 3135 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  D
2621, 25eqeq12i 2309 . . . . . . . . . 10  |-  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  C  =  D
)
2726imbi1i 315 . . . . . . . . 9  |-  ( (
[_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  ( C  =  D  ->  x  =  y )
)
28272ralbii 2582 . . . . . . . 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 ) )
29 nfcsb1v 3126 . . . . . . . . . . . 12  |-  F/_ x [_ y  /  x ]_ C
309, 29nfeq 2439 . . . . . . . . . . 11  |-  F/ x [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
31 nfv 1609 . . . . . . . . . . 11  |-  F/ x  a  =  y
3230, 31nfim 1781 . . . . . . . . . 10  |-  F/ x
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )
33 nfv 1609 . . . . . . . . . 10  |-  F/ y ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  ->  a  =  b )
34 csbeq1 3097 . . . . . . . . . . . 12  |-  ( x  =  a  ->  [_ x  /  x ]_ C  = 
[_ a  /  x ]_ C )
3534eqeq1d 2304 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
) )
36 equequ1 1667 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
3735, 36imbi12d 311 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  (
[_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y ) ) )
38 csbeq1 3097 . . . . . . . . . . . 12  |-  ( y  =  b  ->  [_ y  /  x ]_ C  = 
[_ b  /  x ]_ C )
3938eqeq2d 2307 . . . . . . . . . . 11  |-  ( y  =  b  ->  ( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ b  /  x ]_ C
) )
40 equequ2 1669 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
4139, 40imbi12d 311 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )  <-> 
( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4232, 33, 37, 41rspc2 2902 . . . . . . . . 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 ) ) )
4342com12 27 . . . . . . . 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 ) ) )
4428, 43sylbir 204 . . . . . . 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 ) ) )
45 id 19 . . . . . . . 8  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  ->  a  =  b ) )
46 csbeq1 3097 . . . . . . . 8  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
4745, 46impbid1 194 . . . . . . 7  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  <->  a  =  b ) )
4844, 47syl6 29 . . . . . 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 ) ) )
4948adantl 452 . . . . 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 ) ) )
5020, 49dom2d 6918 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( B  e.  _V  ->  A  ~<_  B ) )
517, 50mpd 14 . . 3  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  A  ~<_  B )
523, 51mtand 640 . 2  |-  ( ph  ->  -.  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
53 ancom 437 . . . . . . 7  |-  ( ( -.  x  =  y  /\  C  =  D )  <->  ( C  =  D  /\  -.  x  =  y ) )
54 df-ne 2461 . . . . . . . 8  |-  ( x  =/=  y  <->  -.  x  =  y )
5554anbi1i 676 . . . . . . 7  |-  ( ( x  =/=  y  /\  C  =  D )  <->  ( -.  x  =  y  /\  C  =  D ) )
56 pm4.61 415 . . . . . . 7  |-  ( -.  ( C  =  D  ->  x  =  y )  <->  ( C  =  D  /\  -.  x  =  y ) )
5753, 55, 563bitr4i 268 . . . . . 6  |-  ( ( x  =/=  y  /\  C  =  D )  <->  -.  ( C  =  D  ->  x  =  y ) )
5857rexbii 2581 . . . . 5  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  E. y  e.  A  -.  ( C  =  D  ->  x  =  y ) )
59 rexnal 2567 . . . . 5  |-  ( E. y  e.  A  -.  ( C  =  D  ->  x  =  y )  <->  -.  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6058, 59bitri 240 . . . 4  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  -. 
A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6160rexbii 2581 . . 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 )
)
62 rexnal 2567 . . 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 ) )
6361, 62bitri 240 . 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 ) )
6452, 63sylibr 203 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   [_csb 3094   class class class wbr 4039    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  fphpdo  27003  pellex  27023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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