MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marypha1 Unicode version

Theorem marypha1 7203
Description: (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pidgeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypotheses
Ref Expression
marypha1.a  |-  ( ph  ->  A  e.  Fin )
marypha1.b  |-  ( ph  ->  B  e.  Fin )
marypha1.c  |-  ( ph  ->  C  C_  ( A  X.  B ) )
marypha1.d  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( C " d ) )
Assertion
Ref Expression
marypha1  |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> B )
Distinct variable groups:    ph, d, f    A, d, f    C, d, f
Allowed substitution hints:    B( f, d)

Proof of Theorem marypha1
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 3646 . . . . 5  |-  ( d  e.  ~P A  -> 
d  C_  A )
2 marypha1.d . . . . 5  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( C " d ) )
31, 2sylan2 460 . . . 4  |-  ( (
ph  /\  d  e.  ~P A )  ->  d  ~<_  ( C " d ) )
43ralrimiva 2639 . . 3  |-  ( ph  ->  A. d  e.  ~P  A d  ~<_  ( C
" d ) )
5 marypha1.c . . . . 5  |-  ( ph  ->  C  C_  ( A  X.  B ) )
6 marypha1.a . . . . . . 7  |-  ( ph  ->  A  e.  Fin )
7 marypha1.b . . . . . . 7  |-  ( ph  ->  B  e.  Fin )
8 xpexg 4816 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B
)  e.  _V )
96, 7, 8syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
10 elpw2g 4190 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  ( C  e.  ~P ( A  X.  B )  <->  C  C_  ( A  X.  B ) ) )
119, 10syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  ~P ( A  X.  B
)  <->  C  C_  ( A  X.  B ) ) )
125, 11mpbird 223 . . . 4  |-  ( ph  ->  C  e.  ~P ( A  X.  B ) )
13 xpeq2 4720 . . . . . . . . 9  |-  ( b  =  B  ->  ( A  X.  b )  =  ( A  X.  B
) )
1413pweqd 3643 . . . . . . . 8  |-  ( b  =  B  ->  ~P ( A  X.  b
)  =  ~P ( A  X.  B ) )
1514raleqdv 2755 . . . . . . 7  |-  ( b  =  B  ->  ( A. c  e.  ~P  ( A  X.  b
) ( A. d  e.  ~P  A d  ~<_  ( c " d )  ->  E. f  e.  ~P  c f : A -1-1-> _V )  <->  A. c  e.  ~P  ( A  X.  B
) ( A. d  e.  ~P  A d  ~<_  ( c " d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) ) )
1615imbi2d 307 . . . . . 6  |-  ( b  =  B  ->  (
( A  e.  Fin  ->  A. c  e.  ~P  ( A  X.  b
) ( A. d  e.  ~P  A d  ~<_  ( c " d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) )  <->  ( A  e.  Fin  ->  A. c  e.  ~P  ( A  X.  B ) ( A. d  e.  ~P  A
d  ~<_  ( c "
d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) ) ) )
17 marypha1lem 7202 . . . . . . 7  |-  ( A  e.  Fin  ->  (
b  e.  Fin  ->  A. c  e.  ~P  ( A  X.  b ) ( A. d  e.  ~P  A d  ~<_  ( c
" d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) ) )
1817com12 27 . . . . . 6  |-  ( b  e.  Fin  ->  ( A  e.  Fin  ->  A. c  e.  ~P  ( A  X.  b ) ( A. d  e.  ~P  A
d  ~<_  ( c "
d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) ) )
1916, 18vtoclga 2862 . . . . 5  |-  ( B  e.  Fin  ->  ( A  e.  Fin  ->  A. c  e.  ~P  ( A  X.  B ) ( A. d  e.  ~P  A
d  ~<_  ( c "
d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) ) )
207, 6, 19sylc 56 . . . 4  |-  ( ph  ->  A. c  e.  ~P  ( A  X.  B
) ( A. d  e.  ~P  A d  ~<_  ( c " d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) )
21 imaeq1 5023 . . . . . . . 8  |-  ( c  =  C  ->  (
c " d )  =  ( C "
d ) )
2221breq2d 4051 . . . . . . 7  |-  ( c  =  C  ->  (
d  ~<_  ( c "
d )  <->  d  ~<_  ( C
" d ) ) )
2322ralbidv 2576 . . . . . 6  |-  ( c  =  C  ->  ( A. d  e.  ~P  A d  ~<_  ( c
" d )  <->  A. d  e.  ~P  A d  ~<_  ( C " d ) ) )
24 pweq 3641 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
2524rexeqdv 2756 . . . . . 6  |-  ( c  =  C  ->  ( E. f  e.  ~P  c f : A -1-1-> _V  <->  E. f  e.  ~P  C
f : A -1-1-> _V ) )
2623, 25imbi12d 311 . . . . 5  |-  ( c  =  C  ->  (
( A. d  e. 
~P  A d  ~<_  ( c " d )  ->  E. f  e.  ~P  c f : A -1-1-> _V )  <->  ( A. d  e.  ~P  A d  ~<_  ( C " d )  ->  E. f  e.  ~P  C f : A -1-1-> _V ) ) )
2726rspcva 2895 . . . 4  |-  ( ( C  e.  ~P ( A  X.  B )  /\  A. c  e.  ~P  ( A  X.  B ) ( A. d  e.  ~P  A d  ~<_  ( c
" d )  ->  E. f  e.  ~P  c f : A -1-1-> _V ) )  ->  ( A. d  e.  ~P  A d  ~<_  ( C
" d )  ->  E. f  e.  ~P  C f : A -1-1-> _V ) )
2812, 20, 27syl2anc 642 . . 3  |-  ( ph  ->  ( A. d  e. 
~P  A d  ~<_  ( C " d )  ->  E. f  e.  ~P  C f : A -1-1-> _V ) )
294, 28mpd 14 . 2  |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> _V )
30 elpwi 3646 . . . . . . 7  |-  ( f  e.  ~P C  -> 
f  C_  C )
3130, 5sylan9ssr 3206 . . . . . 6  |-  ( (
ph  /\  f  e.  ~P C )  ->  f  C_  ( A  X.  B
) )
32 rnss 4923 . . . . . 6  |-  ( f 
C_  ( A  X.  B )  ->  ran  f  C_  ran  ( A  X.  B ) )
3331, 32syl 15 . . . . 5  |-  ( (
ph  /\  f  e.  ~P C )  ->  ran  f  C_  ran  ( A  X.  B ) )
34 rnxpss 5124 . . . . 5  |-  ran  ( A  X.  B )  C_  B
3533, 34syl6ss 3204 . . . 4  |-  ( (
ph  /\  f  e.  ~P C )  ->  ran  f  C_  B )
36 f1ssr 5459 . . . . 5  |-  ( ( f : A -1-1-> _V  /\ 
ran  f  C_  B
)  ->  f : A -1-1-> B )
3736expcom 424 . . . 4  |-  ( ran  f  C_  B  ->  ( f : A -1-1-> _V  ->  f : A -1-1-> B
) )
3835, 37syl 15 . . 3  |-  ( (
ph  /\  f  e.  ~P C )  ->  (
f : A -1-1-> _V  ->  f : A -1-1-> B
) )
3938reximdva 2668 . 2  |-  ( ph  ->  ( E. f  e. 
~P  C f : A -1-1-> _V  ->  E. f  e.  ~P  C f : A -1-1-> B ) )
4029, 39mpd 14 1  |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    X. cxp 4703   ran crn 4706   "cima 4708   -1-1->wf1 5268    ~<_ cdom 6877   Fincfn 6879
This theorem is referenced by:  marypha2  7208
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-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-3or 935  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
  Copyright terms: Public domain W3C validator