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Theorem fphpd 26252
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
StepHypRef Expression
1 domnsym 6941 . . . 4  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
2 fphpd.a . . . 4  |-  ( ph  ->  B  ~<  A )
31, 2nsyl3 113 . . 3  |-  ( ph  ->  -.  A  ~<_  B )
4 relsdom 6824 . . . . . . 7  |-  Rel  ~<
54brrelexi 4703 . . . . . 6  |-  ( B 
~<  A  ->  B  e. 
_V )
62, 5syl 17 . . . . 5  |-  ( ph  ->  B  e.  _V )
76adantr 453 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  B  e.  _V )
8 nfv 1629 . . . . . . . . 9  |-  F/ x
( ph  /\  a  e.  A )
9 nfcsb1v 3074 . . . . . . . . . 10  |-  F/_ x [_ a  /  x ]_ C
109nfel1 2402 . . . . . . . . 9  |-  F/ x [_ a  /  x ]_ C  e.  B
118, 10nfim 1735 . . . . . . . 8  |-  F/ x
( ( ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B
)
12 eleq1 2316 . . . . . . . . . 10  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
1312anbi2d 687 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  a  e.  A ) ) )
14 csbeq1a 3050 . . . . . . . . . 10  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1514eleq1d 2322 . . . . . . . . 9  |-  ( x  =  a  ->  ( C  e.  B  <->  [_ a  /  x ]_ C  e.  B
) )
1613, 15imbi12d 313 . . . . . . . 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 1879 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B )
1918ex 425 . . . . . 6  |-  ( ph  ->  ( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
2019adantr 453 . . . . 5  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
21 csbid 3049 . . . . . . . . . . 11  |-  [_ x  /  x ]_ C  =  C
22 vex 2760 . . . . . . . . . . . 12  |-  y  e. 
_V
23 nfcv 2392 . . . . . . . . . . . 12  |-  F/_ x D
24 fphpd.c . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  D )
2522, 23, 24csbief 3083 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  D
2621, 25eqeq12i 2269 . . . . . . . . . 10  |-  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  C  =  D
)
2726imbi1i 317 . . . . . . . . 9  |-  ( (
[_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  ( C  =  D  ->  x  =  y )
)
28272ralbii 2542 . . . . . . . 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 3074 . . . . . . . . . . . 12  |-  F/_ x [_ y  /  x ]_ C
309, 29nfeq 2399 . . . . . . . . . . 11  |-  F/ x [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
31 nfv 1629 . . . . . . . . . . 11  |-  F/ x  a  =  y
3230, 31nfim 1735 . . . . . . . . . 10  |-  F/ x
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )
33 nfv 1629 . . . . . . . . . 10  |-  F/ y ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  ->  a  =  b )
34 csbeq1 3045 . . . . . . . . . . . 12  |-  ( x  =  a  ->  [_ x  /  x ]_ C  = 
[_ a  /  x ]_ C )
3534eqeq1d 2264 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
) )
36 equequ1 1829 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
3735, 36imbi12d 313 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  (
[_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y ) ) )
38 csbeq1 3045 . . . . . . . . . . . 12  |-  ( y  =  b  ->  [_ y  /  x ]_ C  = 
[_ b  /  x ]_ C )
3938eqeq2d 2267 . . . . . . . . . . 11  |-  ( y  =  b  ->  ( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ b  /  x ]_ C
) )
40 equequ2 1830 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
4139, 40imbi12d 313 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )  <-> 
( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4232, 33, 37, 41rcla42 2857 . . . . . . . . 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 29 . . . . . . . 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 206 . . . . . . 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 21 . . . . . . . 8  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  ->  a  =  b ) )
46 csbeq1 3045 . . . . . . . 8  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
4745, 46impbid1 196 . . . . . . 7  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  <->  a  =  b ) )
4844, 47syl6 31 . . . . . 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 454 . . . . 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 6856 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( B  e.  _V  ->  A  ~<_  B ) )
517, 50mpd 16 . . 3  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  A  ~<_  B )
523, 51mtand 643 . 2  |-  ( ph  ->  -.  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
53 ancom 439 . . . . . . 7  |-  ( ( -.  x  =  y  /\  C  =  D )  <->  ( C  =  D  /\  -.  x  =  y ) )
54 df-ne 2421 . . . . . . . 8  |-  ( x  =/=  y  <->  -.  x  =  y )
5554anbi1i 679 . . . . . . 7  |-  ( ( x  =/=  y  /\  C  =  D )  <->  ( -.  x  =  y  /\  C  =  D ) )
56 pm4.61 417 . . . . . . 7  |-  ( -.  ( C  =  D  ->  x  =  y )  <->  ( C  =  D  /\  -.  x  =  y ) )
5753, 55, 563bitr4i 270 . . . . . 6  |-  ( ( x  =/=  y  /\  C  =  D )  <->  -.  ( C  =  D  ->  x  =  y ) )
5857rexbii 2541 . . . . 5  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  E. y  e.  A  -.  ( C  =  D  ->  x  =  y ) )
59 rexnal 2527 . . . . 5  |-  ( E. y  e.  A  -.  ( C  =  D  ->  x  =  y )  <->  -.  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6058, 59bitri 242 . . . 4  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  -. 
A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6160rexbii 2541 . . 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 2527 . . 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 242 . 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 205 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   _Vcvv 2757   [_csb 3042   class class class wbr 3983    ~<_ cdom 6815    ~< csdm 6816
This theorem is referenced by:  fphpdo  26253  pellex  26273
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820
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