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Theorem marypha1 7441
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 3809 . . . . 5  |-  ( d  e.  ~P A  -> 
d  C_  A )
2 marypha1.d . . . . 5  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( C " d ) )
31, 2sylan2 462 . . . 4  |-  ( (
ph  /\  d  e.  ~P A )  ->  d  ~<_  ( C " d ) )
43ralrimiva 2791 . . 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 4991 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B
)  e.  _V )
96, 7, 8syl2anc 644 . . . . . 6  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
10 elpw2g 4365 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  ( C  e.  ~P ( A  X.  B )  <->  C  C_  ( A  X.  B ) ) )
119, 10syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  ~P ( A  X.  B
)  <->  C  C_  ( A  X.  B ) ) )
125, 11mpbird 225 . . . 4  |-  ( ph  ->  C  e.  ~P ( A  X.  B ) )
13 xpeq2 4895 . . . . . . . . 9  |-  ( b  =  B  ->  ( A  X.  b )  =  ( A  X.  B
) )
1413pweqd 3806 . . . . . . . 8  |-  ( b  =  B  ->  ~P ( A  X.  b
)  =  ~P ( A  X.  B ) )
1514raleqdv 2912 . . . . . . 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 309 . . . . . 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 7440 . . . . . . 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 30 . . . . . 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 3019 . . . . 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 59 . . . 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 5200 . . . . . . . 8  |-  ( c  =  C  ->  (
c " d )  =  ( C "
d ) )
2221breq2d 4226 . . . . . . 7  |-  ( c  =  C  ->  (
d  ~<_  ( c "
d )  <->  d  ~<_  ( C
" d ) ) )
2322ralbidv 2727 . . . . . 6  |-  ( c  =  C  ->  ( A. d  e.  ~P  A d  ~<_  ( c
" d )  <->  A. d  e.  ~P  A d  ~<_  ( C " d ) ) )
24 pweq 3804 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
2524rexeqdv 2913 . . . . . 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 313 . . . . 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 3052 . . . 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 644 . . 3  |-  ( ph  ->  ( A. d  e. 
~P  A d  ~<_  ( C " d )  ->  E. f  e.  ~P  C f : A -1-1-> _V ) )
294, 28mpd 15 . 2  |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> _V )
30 elpwi 3809 . . . . . . 7  |-  ( f  e.  ~P C  -> 
f  C_  C )
3130, 5sylan9ssr 3364 . . . . . 6  |-  ( (
ph  /\  f  e.  ~P C )  ->  f  C_  ( A  X.  B
) )
32 rnss 5100 . . . . . 6  |-  ( f 
C_  ( A  X.  B )  ->  ran  f  C_  ran  ( A  X.  B ) )
3331, 32syl 16 . . . . 5  |-  ( (
ph  /\  f  e.  ~P C )  ->  ran  f  C_  ran  ( A  X.  B ) )
34 rnxpss 5303 . . . . 5  |-  ran  ( A  X.  B )  C_  B
3533, 34syl6ss 3362 . . . 4  |-  ( (
ph  /\  f  e.  ~P C )  ->  ran  f  C_  B )
36 f1ssr 5647 . . . . 5  |-  ( ( f : A -1-1-> _V  /\ 
ran  f  C_  B
)  ->  f : A -1-1-> B )
3736expcom 426 . . . 4  |-  ( ran  f  C_  B  ->  ( f : A -1-1-> _V  ->  f : A -1-1-> B
) )
3835, 37syl 16 . . 3  |-  ( (
ph  /\  f  e.  ~P C )  ->  (
f : A -1-1-> _V  ->  f : A -1-1-> B
) )
3938reximdva 2820 . 2  |-  ( ph  ->  ( E. f  e. 
~P  C f : A -1-1-> _V  ->  E. f  e.  ~P  C f : A -1-1-> B ) )
4029, 39mpd 15 1  |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214    X. cxp 4878   ran crn 4881   "cima 4883   -1-1->wf1 5453    ~<_ cdom 7109   Fincfn 7111
This theorem is referenced by:  marypha2  7446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115
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