Theorem fphpd 37380

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.

Step	Hyp	Ref	Expression
1		domnsym 8086	. . . 4 |- ( A ~<_ B -> -. B ~< A )
2		fphpd.a	. . . 4 |- ( ph -> B ~< A )
3	1, 2	nsyl3 133	. . 3 |- ( ph -> -. A ~<_ B )
4		relsdom 7962	. . . . . . 7 |- Rel ~<
5	4	brrelexi 5158	. . . . . 6 |- ( B ~< A -> B e. _V )
6	2, 5	syl 17	. . . . 5 |- ( ph -> B e. _V )
7	6	adantr 481	. . . 4 |- ( ( ph /\ A. x e. A A. y e. A ( C = D -> x = y ) ) -> B e. _V )
8		nfv 1843	. . . . . . . . 9 |- F/ x ( ph /\ a e. A )
9		nfcsb1v 3549	. . . . . . . . . 10 |- F/_ x [_ a / x ]_ C
10	9	nfel1 2779	. . . . . . . . 9 |- F/ x [_ a / x ]_ C e. B
11	8, 10	nfim 1825	. . . . . . . 8 |- F/ x ( ( ph /\ a e. A ) -> [_ a / x ]_ C e. B )
12		eleq1 2689	. . . . . . . . . 10 |- ( x = a -> ( x e. A <-> a e. A ) )
13	12	anbi2d 740	. . . . . . . . 9 |- ( x = a -> ( ( ph /\ x e. A ) <-> ( ph /\ a e. A ) ) )
14		csbeq1a 3542	. . . . . . . . . 10 |- ( x = a -> C = [_ a / x ]_ C )
15	14	eleq1d 2686	. . . . . . . . 9 |- ( x = a -> ( C e. B <-> [_ a / x ]_ C e. B ) )
16	13, 15	imbi12d 334	. . . . . . . 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 )
18	11, 16, 17	chvar 2262	. . . . . . 7 |- ( ( ph /\ a e. A ) -> [_ a / x ]_ C e. B )
19	18	ex 450	. . . . . 6 |- ( ph -> ( a e. A -> [_ a / x ]_ C e. B ) )
20	19	adantr 481	. . . . 5 |- ( ( ph /\ A. x e. A A. y e. A ( C = D -> x = y ) ) -> ( a e. A -> [_ a / x ]_ C e. B ) )
21		csbid 3541	. . . . . . . . . . 11 |- [_ x / x ]_ C = C
22		vex 3203	. . . . . . . . . . . 12 |- y e. _V
23		fphpd.c	. . . . . . . . . . . 12 |- ( x = y -> C = D )
24	22, 23	csbie 3559	. . . . . . . . . . 11 |- [_ y / x ]_ C = D
25	21, 24	eqeq12i 2636	. . . . . . . . . 10 |- ( [_ x / x ]_ C = [_ y / x ]_ C <-> C = D )
26	25	imbi1i 339	. . . . . . . . 9 |- ( ( [_ x / x ]_ C = [_ y / x ]_ C -> x = y ) <-> ( C = D -> x = y ) )
27	26	2ralbii 2981	. . . . . . . 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 ) )
28		nfcsb1v 3549	. . . . . . . . . . . 12 |- F/_ x [_ y / x ]_ C
29	9, 28	nfeq 2776	. . . . . . . . . . 11 |- F/ x [_ a / x ]_ C = [_ y / x ]_ C
30		nfv 1843	. . . . . . . . . . 11 |- F/ x a = y
31	29, 30	nfim 1825	. . . . . . . . . 10 |- F/ x ( [_ a / x ]_ C = [_ y / x ]_ C -> a = y )
32		nfv 1843	. . . . . . . . . 10 |- F/ y ( [_ a / x ]_ C = [_ b / x ]_ C -> a = b )
33		csbeq1 3536	. . . . . . . . . . . 12 |- ( x = a -> [_ x / x ]_ C = [_ a / x ]_ C )
34	33	eqeq1d 2624	. . . . . . . . . . 11 |- ( x = a -> ( [_ x / x ]_ C = [_ y / x ]_ C <-> [_ a / x ]_ C = [_ y / x ]_ C ) )
35		equequ1 1952	. . . . . . . . . . 11 |- ( x = a -> ( x = y <-> a = y ) )
36	34, 35	imbi12d 334	. . . . . . . . . 10 |- ( x = a -> ( ( [_ x / x ]_ C = [_ y / x ]_ C -> x = y ) <-> ( [_ a / x ]_ C = [_ y / x ]_ C -> a = y ) ) )
37		csbeq1 3536	. . . . . . . . . . . 12 |- ( y = b -> [_ y / x ]_ C = [_ b / x ]_ C )
38	37	eqeq2d 2632	. . . . . . . . . . 11 |- ( y = b -> ( [_ a / x ]_ C = [_ y / x ]_ C <-> [_ a / x ]_ C = [_ b / x ]_ C ) )
39		equequ2 1953	. . . . . . . . . . 11 |- ( y = b -> ( a = y <-> a = b ) )
40	38, 39	imbi12d 334	. . . . . . . . . 10 |- ( y = b -> ( ( [_ a / x ]_ C = [_ y / x ]_ C -> a = y ) <-> ( [_ a / x ]_ C = [_ b / x ]_ C -> a = b ) ) )
41	31, 32, 36, 40	rspc2 3320	. . . . . . . . 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 ) ) )
42	41	com12 32	. . . . . . . 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 ) ) )
43	27, 42	sylbir 225	. . . . . . 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 ) ) )
44		id 22	. . . . . . . 8 |- ( ( [_ a / x ]_ C = [_ b / x ]_ C -> a = b ) -> ( [_ a / x ]_ C = [_ b / x ]_ C -> a = b ) )
45		csbeq1 3536	. . . . . . . 8 |- ( a = b -> [_ a / x ]_ C = [_ b / x ]_ C )
46	44, 45	impbid1 215	. . . . . . 7 |- ( ( [_ a / x ]_ C = [_ b / x ]_ C -> a = b ) -> ( [_ a / x ]_ C = [_ b / x ]_ C <-> a = b ) )
47	43, 46	syl6 35	. . . . . 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 ) ) )
48	47	adantl 482	. . . . 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 ) ) )
49	20, 48	dom2d 7996	. . . 4 |- ( ( ph /\ A. x e. A A. y e. A ( C = D -> x = y ) ) -> ( B e. _V -> A ~<_ B ) )
50	7, 49	mpd 15	. . 3 |- ( ( ph /\ A. x e. A A. y e. A ( C = D -> x = y ) ) -> A ~<_ B )
51	3, 50	mtand 691	. 2 |- ( ph -> -. A. x e. A A. y e. A ( C = D -> x = y ) )
52		ancom 466	. . . . . . 7 |- ( ( -. x = y /\ C = D ) <-> ( C = D /\ -. x = y ) )
53		df-ne 2795	. . . . . . . 8 |- ( x =/= y <-> -. x = y )
54	53	anbi1i 731	. . . . . . 7 |- ( ( x =/= y /\ C = D ) <-> ( -. x = y /\ C = D ) )
55		pm4.61 442	. . . . . . 7 |- ( -. ( C = D -> x = y ) <-> ( C = D /\ -. x = y ) )
56	52, 54, 55	3bitr4i 292	. . . . . 6 |- ( ( x =/= y /\ C = D ) <-> -. ( C = D -> x = y ) )
57	56	rexbii 3041	. . . . 5 |- ( E. y e. A ( x =/= y /\ C = D ) <-> E. y e. A -. ( C = D -> x = y ) )
58		rexnal 2995	. . . . 5 |- ( E. y e. A -. ( C = D -> x = y ) <-> -. A. y e. A ( C = D -> x = y ) )
59	57, 58	bitri 264	. . . 4 |- ( E. y e. A ( x =/= y /\ C = D ) <-> -. A. y e. A ( C = D -> x = y ) )
60	59	rexbii 3041	. . 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 ) )
61		rexnal 2995	. . 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 ) )
62	60, 61	bitri 264	. 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 ) )
63	51, 62	sylibr 224	1 |- ( ph -> E. x e. A E. y e. A ( x =/= y /\ C = D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   [_csb 3533   class class class wbr 4653   ~<_ cdom 7953   ~< csdm 7954

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-rep 4771  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-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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  fphpdo  37381  pellex  37399