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Theorem fphpd 29296
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 7540 . . . 4  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
2 fphpd.a . . . 4  |-  ( ph  ->  B  ~<  A )
31, 2nsyl3 119 . . 3  |-  ( ph  ->  -.  A  ~<_  B )
4 relsdom 7420 . . . . . . 7  |-  Rel  ~<
54brrelexi 4980 . . . . . 6  |-  ( B 
~<  A  ->  B  e. 
_V )
62, 5syl 16 . . . . 5  |-  ( ph  ->  B  e.  _V )
76adantr 465 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  B  e.  _V )
8 nfv 1674 . . . . . . . . 9  |-  F/ x
( ph  /\  a  e.  A )
9 nfcsb1v 3405 . . . . . . . . . 10  |-  F/_ x [_ a  /  x ]_ C
109nfel1 2628 . . . . . . . . 9  |-  F/ x [_ a  /  x ]_ C  e.  B
118, 10nfim 1855 . . . . . . . 8  |-  F/ x
( ( ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B
)
12 eleq1 2523 . . . . . . . . . 10  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
1312anbi2d 703 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  a  e.  A ) ) )
14 csbeq1a 3398 . . . . . . . . . 10  |-  ( x  =  a  ->  C  =  [_ a  /  x ]_ C )
1514eleq1d 2520 . . . . . . . . 9  |-  ( x  =  a  ->  ( C  e.  B  <->  [_ a  /  x ]_ C  e.  B
) )
1613, 15imbi12d 320 . . . . . . . 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 1966 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  [_ a  /  x ]_ C  e.  B )
1918ex 434 . . . . . 6  |-  ( ph  ->  ( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( a  e.  A  ->  [_ a  /  x ]_ C  e.  B
) )
21 csbid 3397 . . . . . . . . . . 11  |-  [_ x  /  x ]_ C  =  C
22 vex 3074 . . . . . . . . . . . 12  |-  y  e. 
_V
23 nfcv 2613 . . . . . . . . . . . 12  |-  F/_ x D
24 fphpd.c . . . . . . . . . . . 12  |-  ( x  =  y  ->  C  =  D )
2522, 23, 24csbief 3414 . . . . . . . . . . 11  |-  [_ y  /  x ]_ C  =  D
2621, 25eqeq12i 2471 . . . . . . . . . 10  |-  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  C  =  D
)
2726imbi1i 325 . . . . . . . . 9  |-  ( (
[_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  ( C  =  D  ->  x  =  y )
)
28272ralbii 2835 . . . . . . . 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 3405 . . . . . . . . . . . 12  |-  F/_ x [_ y  /  x ]_ C
309, 29nfeq 2623 . . . . . . . . . . 11  |-  F/ x [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
31 nfv 1674 . . . . . . . . . . 11  |-  F/ x  a  =  y
3230, 31nfim 1855 . . . . . . . . . 10  |-  F/ x
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )
33 nfv 1674 . . . . . . . . . 10  |-  F/ y ( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  ->  a  =  b )
34 csbeq1 3392 . . . . . . . . . . . 12  |-  ( x  =  a  ->  [_ x  /  x ]_ C  = 
[_ a  /  x ]_ C )
3534eqeq1d 2453 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ y  /  x ]_ C
) )
36 equequ1 1738 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
3735, 36imbi12d 320 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( [_ x  /  x ]_ C  =  [_ y  /  x ]_ C  ->  x  =  y )  <->  (
[_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y ) ) )
38 csbeq1 3392 . . . . . . . . . . . 12  |-  ( y  =  b  ->  [_ y  /  x ]_ C  = 
[_ b  /  x ]_ C )
3938eqeq2d 2465 . . . . . . . . . . 11  |-  ( y  =  b  ->  ( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  <->  [_ a  /  x ]_ C  =  [_ b  /  x ]_ C
) )
40 equequ2 1739 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
4139, 40imbi12d 320 . . . . . . . . . 10  |-  ( y  =  b  ->  (
( [_ a  /  x ]_ C  =  [_ y  /  x ]_ C  -> 
a  =  y )  <-> 
( [_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b ) ) )
4232, 33, 37, 41rspc2 3178 . . . . . . . . 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 31 . . . . . . . 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 213 . . . . . . 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 22 . . . . . . . 8  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  ->  a  =  b ) )
46 csbeq1 3392 . . . . . . . 8  |-  ( a  =  b  ->  [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C )
4745, 46impbid1 203 . . . . . . 7  |-  ( (
[_ a  /  x ]_ C  =  [_ b  /  x ]_ C  -> 
a  =  b )  ->  ( [_ a  /  x ]_ C  = 
[_ b  /  x ]_ C  <->  a  =  b ) )
4844, 47syl6 33 . . . . . 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 466 . . . . 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 7453 . . . 4  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  -> 
( B  e.  _V  ->  A  ~<_  B ) )
517, 50mpd 15 . . 3  |-  ( (
ph  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )  ->  A  ~<_  B )
523, 51mtand 659 . 2  |-  ( ph  ->  -.  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
53 ancom 450 . . . . . . 7  |-  ( ( -.  x  =  y  /\  C  =  D )  <->  ( C  =  D  /\  -.  x  =  y ) )
54 df-ne 2646 . . . . . . . 8  |-  ( x  =/=  y  <->  -.  x  =  y )
5554anbi1i 695 . . . . . . 7  |-  ( ( x  =/=  y  /\  C  =  D )  <->  ( -.  x  =  y  /\  C  =  D ) )
56 pm4.61 426 . . . . . . 7  |-  ( -.  ( C  =  D  ->  x  =  y )  <->  ( C  =  D  /\  -.  x  =  y ) )
5753, 55, 563bitr4i 277 . . . . . 6  |-  ( ( x  =/=  y  /\  C  =  D )  <->  -.  ( C  =  D  ->  x  =  y ) )
5857rexbii 2859 . . . . 5  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  E. y  e.  A  -.  ( C  =  D  ->  x  =  y ) )
59 rexnal 2847 . . . . 5  |-  ( E. y  e.  A  -.  ( C  =  D  ->  x  =  y )  <->  -.  A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6058, 59bitri 249 . . . 4  |-  ( E. y  e.  A  ( x  =/=  y  /\  C  =  D )  <->  -. 
A. y  e.  A  ( C  =  D  ->  x  =  y ) )
6160rexbii 2859 . . 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 2847 . . 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 249 . 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 212 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 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   _Vcvv 3071   [_csb 3389   class class class wbr 4393    ~<_ cdom 7411    ~< csdm 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416
This theorem is referenced by:  fphpdo  29297  pellex  29317
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