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Theorem marypha1 7689
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 pigeonhole 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 3874 . . . . 5  |-  ( d  e.  ~P A  -> 
d  C_  A )
2 marypha1.d . . . . 5  |-  ( (
ph  /\  d  C_  A )  ->  d  ~<_  ( C " d ) )
31, 2sylan2 474 . . . 4  |-  ( (
ph  /\  d  e.  ~P A )  ->  d  ~<_  ( C " d ) )
43ralrimiva 2804 . . 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 6512 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B
)  e.  _V )
96, 7, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
10 elpw2g 4460 . . . . . 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 232 . . . 4  |-  ( ph  ->  C  e.  ~P ( A  X.  B ) )
13 xpeq2 4860 . . . . . . . . 9  |-  ( b  =  B  ->  ( A  X.  b )  =  ( A  X.  B
) )
1413pweqd 3870 . . . . . . . 8  |-  ( b  =  B  ->  ~P ( A  X.  b
)  =  ~P ( A  X.  B ) )
1514raleqdv 2928 . . . . . . 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 316 . . . . . 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 7688 . . . . . . 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 31 . . . . . 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 3041 . . . . 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 60 . . . 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 5169 . . . . . . . 8  |-  ( c  =  C  ->  (
c " d )  =  ( C "
d ) )
2221breq2d 4309 . . . . . . 7  |-  ( c  =  C  ->  (
d  ~<_  ( c "
d )  <->  d  ~<_  ( C
" d ) ) )
2322ralbidv 2740 . . . . . 6  |-  ( c  =  C  ->  ( A. d  e.  ~P  A d  ~<_  ( c
" d )  <->  A. d  e.  ~P  A d  ~<_  ( C " d ) ) )
24 pweq 3868 . . . . . . 7  |-  ( c  =  C  ->  ~P c  =  ~P C
)
2524rexeqdv 2929 . . . . . 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 320 . . . . 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 3076 . . . 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 661 . . 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 3874 . . . . . . 7  |-  ( f  e.  ~P C  -> 
f  C_  C )
3130, 5sylan9ssr 3375 . . . . . 6  |-  ( (
ph  /\  f  e.  ~P C )  ->  f  C_  ( A  X.  B
) )
32 rnss 5073 . . . . . 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 5275 . . . . 5  |-  ran  ( A  X.  B )  C_  B
3533, 34syl6ss 3373 . . . 4  |-  ( (
ph  /\  f  e.  ~P C )  ->  ran  f  C_  B )
36 f1ssr 5617 . . . . 5  |-  ( ( f : A -1-1-> _V  /\ 
ran  f  C_  B
)  ->  f : A -1-1-> B )
3736expcom 435 . . . 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 2833 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   class class class wbr 4297    X. cxp 4843   ran crn 4846   "cima 4848   -1-1->wf1 5420    ~<_ cdom 7313   Fincfn 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319
This theorem is referenced by:  marypha2  7694
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