Theorem fidomdm 8243

Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
fidomdm	|- ( F e. Fin -> dom F ~<_ F )

Proof of Theorem fidomdm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmresv 5593	. 2 |- dom ( F |` _V ) = dom F
2		finresfin 8186	. . . 4 |- ( F e. Fin -> ( F |` _V ) e. Fin )
3		fvex 6201	. . . . . . 7 |- ( 1st ` x ) e. _V
4		eqid 2622	. . . . . . 7 |- ( x e. ( F |` _V ) |-> ( 1st ` x ) ) = ( x e. ( F |` _V ) |-> ( 1st ` x ) )
5	3, 4	fnmpti 6022	. . . . . 6 |- ( x e. ( F |` _V ) |-> ( 1st ` x ) ) Fn ( F |` _V )
6		dffn4 6121	. . . . . 6 |- ( ( x e. ( F |` _V ) |-> ( 1st ` x ) ) Fn ( F |` _V ) <-> ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> ran ( x e. ( F |` _V ) |-> ( 1st ` x ) ) )
7	5, 6	mpbi 220	. . . . 5 |- ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> ran ( x e. ( F |` _V ) |-> ( 1st ` x ) )
8		relres 5426	. . . . . 6 |- Rel ( F |` _V )
9		reldm 7219	. . . . . 6 |- ( Rel ( F |` _V ) -> dom ( F |` _V ) = ran ( x e. ( F |` _V ) |-> ( 1st ` x ) ) )
10		foeq3 6113	. . . . . 6 |- ( dom ( F |` _V ) = ran ( x e. ( F |` _V ) |-> ( 1st ` x ) ) -> ( ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> dom ( F |` _V ) <-> ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> ran ( x e. ( F |` _V ) |-> ( 1st ` x ) ) ) )
11	8, 9, 10	mp2b 10	. . . . 5 |- ( ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> dom ( F |` _V ) <-> ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> ran ( x e. ( F |` _V ) |-> ( 1st ` x ) ) )
12	7, 11	mpbir 221	. . . 4 |- ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> dom ( F |` _V )
13		fodomfi 8239	. . . 4 |- ( ( ( F |` _V ) e. Fin /\ ( x e. ( F |` _V ) |-> ( 1st ` x ) ) : ( F |` _V ) -onto-> dom ( F |` _V ) ) -> dom ( F |` _V ) ~<_ ( F |` _V ) )
14	2, 12, 13	sylancl 694	. . 3 |- ( F e. Fin -> dom ( F |` _V ) ~<_ ( F |` _V ) )
15		resss 5422	. . . 4 |- ( F |` _V ) C_ F
16		ssdomg 8001	. . . 4 |- ( F e. Fin -> ( ( F |` _V ) C_ F -> ( F |` _V ) ~<_ F ) )
17	15, 16	mpi 20	. . 3 |- ( F e. Fin -> ( F |` _V ) ~<_ F )
18		domtr 8009	. . 3 |- ( ( dom ( F |` _V ) ~<_ ( F |` _V ) /\ ( F |` _V ) ~<_ F ) -> dom ( F |` _V ) ~<_ F )
19	14, 17, 18	syl2anc 693	. 2 |- ( F e. Fin -> dom ( F |` _V ) ~<_ F )
20	1, 19	syl5eqbrr 4689	1 |- ( F e. Fin -> dom F ~<_ F )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Rel wrel 5119   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   ~<_ 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-reu 2919  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-mpt 4730  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-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959

This theorem is referenced by:  dmfi  8244  hashfun  13224