Theorem marypha1 8340

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.

Step	Hyp	Ref	Expression
1		elpwi 4168	. . . . 5 |- ( d e. ~P A -> d C_ A )
2		marypha1.d	. . . . 5 |- ( ( ph /\ d C_ A ) -> d ~<_ ( C " d ) )
3	1, 2	sylan2 491	. . . 4 |- ( ( ph /\ d e. ~P A ) -> d ~<_ ( C " d ) )
4	3	ralrimiva 2966	. . 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 6960	. . . . . . 7 |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. _V )
9	6, 7, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( A X. B ) e. _V )
10		elpw2g 4827	. . . . . 6 |- ( ( A X. B ) e. _V -> ( C e. ~P ( A X. B ) <-> C C_ ( A X. B ) ) )
11	9, 10	syl 17	. . . . 5 |- ( ph -> ( C e. ~P ( A X. B ) <-> C C_ ( A X. B ) ) )
12	5, 11	mpbird 247	. . . 4 |- ( ph -> C e. ~P ( A X. B ) )
13		xpeq2 5129	. . . . . . . . 9 |- ( b = B -> ( A X. b ) = ( A X. B ) )
14	13	pweqd 4163	. . . . . . . 8 |- ( b = B -> ~P ( A X. b ) = ~P ( A X. B ) )
15	14	raleqdv 3144	. . . . . . 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 ) ) )
16	15	imbi2d 330	. . . . . 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 8339	. . . . . . 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 ) ) )
18	17	com12 32	. . . . . 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 ) ) )
19	16, 18	vtoclga 3272	. . . . 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 ) ) )
20	7, 6, 19	sylc 65	. . . 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 5461	. . . . . . . 8 |- ( c = C -> ( c " d ) = ( C " d ) )
22	21	breq2d 4665	. . . . . . 7 |- ( c = C -> ( d ~<_ ( c " d ) <-> d ~<_ ( C " d ) ) )
23	22	ralbidv 2986	. . . . . 6 |- ( c = C -> ( A. d e. ~P A d ~<_ ( c " d ) <-> A. d e. ~P A d ~<_ ( C " d ) ) )
24		pweq 4161	. . . . . . 7 |- ( c = C -> ~P c = ~P C )
25	24	rexeqdv 3145	. . . . . 6 |- ( c = C -> ( E. f e. ~P c f : A -1-1-> _V <-> E. f e. ~P C f : A -1-1-> _V ) )
26	23, 25	imbi12d 334	. . . . 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 ) ) )
27	26	rspcva 3307	. . . 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 ) )
28	12, 20, 27	syl2anc 693	. . 3 |- ( ph -> ( A. d e. ~P A d ~<_ ( C " d ) -> E. f e. ~P C f : A -1-1-> _V ) )
29	4, 28	mpd 15	. 2 |- ( ph -> E. f e. ~P C f : A -1-1-> _V )
30		elpwi 4168	. . . . . . 7 |- ( f e. ~P C -> f C_ C )
31	30, 5	sylan9ssr 3617	. . . . . 6 |- ( ( ph /\ f e. ~P C ) -> f C_ ( A X. B ) )
32		rnss 5354	. . . . . 6 |- ( f C_ ( A X. B ) -> ran f C_ ran ( A X. B ) )
33	31, 32	syl 17	. . . . 5 |- ( ( ph /\ f e. ~P C ) -> ran f C_ ran ( A X. B ) )
34		rnxpss 5566	. . . . 5 |- ran ( A X. B ) C_ B
35	33, 34	syl6ss 3615	. . . 4 |- ( ( ph /\ f e. ~P C ) -> ran f C_ B )
36		f1ssr 6107	. . . . 5 |- ( ( f : A -1-1-> _V /\ ran f C_ B ) -> f : A -1-1-> B )
37	36	expcom 451	. . . 4 |- ( ran f C_ B -> ( f : A -1-1-> _V -> f : A -1-1-> B ) )
38	35, 37	syl 17	. . 3 |- ( ( ph /\ f e. ~P C ) -> ( f : A -1-1-> _V -> f : A -1-1-> B ) )
39	38	reximdva 3017	. 2 |- ( ph -> ( E. f e. ~P C f : A -1-1-> _V -> E. f e. ~P C f : A -1-1-> B ) )
40	29, 39	mpd 15	1 |- ( ph -> E. f e. ~P C f : A -1-1-> B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112   ran crn 5115   "cima 5117   -1-1->wf1 5885   ~<_ cdom 7953   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  marypha2  8345