Theorem fodomfi 7788

Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8891 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
fodomfi	|- ( ( A e. Fin /\ F : A -onto-> B ) -> B ~<_ A )

Proof of Theorem fodomfi

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foima 5791	. . 3 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	adantl 466	. 2 |- ( ( A e. Fin /\ F : A -onto-> B ) -> ( F " A ) = B )
3		fofn 5788	. . . 4 |- ( F : A -onto-> B -> F Fn A )
4		imaeq2 5324	. . . . . . . 8 |- ( x = (/) -> ( F " x ) = ( F " (/) ) )
5		ima0 5343	. . . . . . . 8 |- ( F " (/) ) = (/)
6	4, 5	syl6eq 2517	. . . . . . 7 |- ( x = (/) -> ( F " x ) = (/) )
7		id 22	. . . . . . 7 |- ( x = (/) -> x = (/) )
8	6, 7	breq12d 4453	. . . . . 6 |- ( x = (/) -> ( ( F " x ) ~<_ x <-> (/) ~<_ (/) ) )
9	8	imbi2d 316	. . . . 5 |- ( x = (/) -> ( ( F Fn A -> ( F " x ) ~<_ x ) <-> ( F Fn A -> (/) ~<_ (/) ) ) )
10		imaeq2 5324	. . . . . . 7 |- ( x = y -> ( F " x ) = ( F " y ) )
11		id 22	. . . . . . 7 |- ( x = y -> x = y )
12	10, 11	breq12d 4453	. . . . . 6 |- ( x = y -> ( ( F " x ) ~<_ x <-> ( F " y ) ~<_ y ) )
13	12	imbi2d 316	. . . . 5 |- ( x = y -> ( ( F Fn A -> ( F " x ) ~<_ x ) <-> ( F Fn A -> ( F " y ) ~<_ y ) ) )
14		imaeq2 5324	. . . . . . 7 |- ( x = ( y u. { z } ) -> ( F " x ) = ( F " ( y u. { z } ) ) )
15		id 22	. . . . . . 7 |- ( x = ( y u. { z } ) -> x = ( y u. { z } ) )
16	14, 15	breq12d 4453	. . . . . 6 |- ( x = ( y u. { z } ) -> ( ( F " x ) ~<_ x <-> ( F " ( y u. { z } ) ) ~<_ ( y u. { z } ) ) )
17	16	imbi2d 316	. . . . 5 |- ( x = ( y u. { z } ) -> ( ( F Fn A -> ( F " x ) ~<_ x ) <-> ( F Fn A -> ( F " ( y u. { z } ) ) ~<_ ( y u. { z } ) ) ) )
18		imaeq2 5324	. . . . . . 7 |- ( x = A -> ( F " x ) = ( F " A ) )
19		id 22	. . . . . . 7 |- ( x = A -> x = A )
20	18, 19	breq12d 4453	. . . . . 6 |- ( x = A -> ( ( F " x ) ~<_ x <-> ( F " A ) ~<_ A ) )
21	20	imbi2d 316	. . . . 5 |- ( x = A -> ( ( F Fn A -> ( F " x ) ~<_ x ) <-> ( F Fn A -> ( F " A ) ~<_ A ) ) )
22		0ex 4570	. . . . . . 7 |- (/) e. _V
23	22	0dom 7637	. . . . . 6 |- (/) ~<_ (/)
24	23	a1i 11	. . . . 5 |- ( F Fn A -> (/) ~<_ (/) )
25		fnfun 5669	. . . . . . . . . . . . . . 15 |- ( F Fn A -> Fun F )
26	25	ad2antrl 727	. . . . . . . . . . . . . 14 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> Fun F )
27		funressn 6065	. . . . . . . . . . . . . 14 |- ( Fun F -> ( F |` { z } ) C_ { <. z , ( F ` z ) >. } )
28		rnss 5222	. . . . . . . . . . . . . 14 |- ( ( F |` { z } ) C_ { <. z , ( F ` z ) >. } -> ran ( F |` { z } ) C_ ran { <. z , ( F ` z ) >. } )
29	26, 27, 28	3syl 20	. . . . . . . . . . . . 13 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ran ( F |` { z } ) C_ ran { <. z , ( F ` z ) >. } )
30		df-ima 5005	. . . . . . . . . . . . 13 |- ( F " { z } ) = ran ( F |` { z } )
31		vex 3109	. . . . . . . . . . . . . . 15 |- z e. _V
32	31	rnsnop 5480	. . . . . . . . . . . . . 14 |- ran { <. z , ( F ` z ) >. } = { ( F ` z ) }
33	32	eqcomi 2473	. . . . . . . . . . . . 13 |- { ( F ` z ) } = ran { <. z , ( F ` z ) >. }
34	29, 30, 33	3sstr4g 3538	. . . . . . . . . . . 12 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " { z } ) C_ { ( F ` z ) } )
35		snex 4681	. . . . . . . . . . . 12 |- { ( F ` z ) } e. _V
36		ssexg 4586	. . . . . . . . . . . 12 |- ( ( ( F " { z } ) C_ { ( F ` z ) } /\ { ( F ` z ) } e. _V ) -> ( F " { z } ) e. _V )
37	34, 35, 36	sylancl 662	. . . . . . . . . . 11 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " { z } ) e. _V )
38		fvi 5915	. . . . . . . . . . 11 |- ( ( F " { z } ) e. _V -> ( _I ` ( F " { z } ) ) = ( F " { z } ) )
39	37, 38	syl 16	. . . . . . . . . 10 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( _I ` ( F " { z } ) ) = ( F " { z } ) )
40	39	uneq2d 3651	. . . . . . . . 9 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( ( F " y ) u. ( _I ` ( F " { z } ) ) ) = ( ( F " y ) u. ( F " { z } ) ) )
41		imaundi 5409	. . . . . . . . 9 |- ( F " ( y u. { z } ) ) = ( ( F " y ) u. ( F " { z } ) )
42	40, 41	syl6eqr 2519	. . . . . . . 8 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( ( F " y ) u. ( _I ` ( F " { z } ) ) ) = ( F " ( y u. { z } ) ) )
43		simprr 756	. . . . . . . . 9 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " y ) ~<_ y )
44		ssdomg 7551	. . . . . . . . . . . 12 |- ( { ( F ` z ) } e. _V -> ( ( F " { z } ) C_ { ( F ` z ) } -> ( F " { z } ) ~<_ { ( F ` z ) } ) )
45	35, 34, 44	mpsyl 63	. . . . . . . . . . 11 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " { z } ) ~<_ { ( F ` z ) } )
46		fvex 5867	. . . . . . . . . . . . 13 |- ( F ` z ) e. _V
47	46	ensn1 7569	. . . . . . . . . . . 12 |- { ( F ` z ) } ~~ 1o
48	31	ensn1 7569	. . . . . . . . . . . 12 |- { z } ~~ 1o
49	47, 48	entr4i 7562	. . . . . . . . . . 11 |- { ( F ` z ) } ~~ { z }
50		domentr 7564	. . . . . . . . . . 11 |- ( ( ( F " { z } ) ~<_ { ( F ` z ) } /\ { ( F ` z ) } ~~ { z } ) -> ( F " { z } ) ~<_ { z } )
51	45, 49, 50	sylancl 662	. . . . . . . . . 10 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " { z } ) ~<_ { z } )
52	39, 51	eqbrtrd 4460	. . . . . . . . 9 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( _I ` ( F " { z } ) ) ~<_ { z } )
53		simplr 754	. . . . . . . . . 10 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> -. z e. y )
54		disjsn 4081	. . . . . . . . . 10 |- ( ( y i^i { z } ) = (/) <-> -. z e. y )
55	53, 54	sylibr 212	. . . . . . . . 9 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( y i^i { z } ) = (/) )
56		undom 7595	. . . . . . . . 9 |- ( ( ( ( F " y ) ~<_ y /\ ( _I ` ( F " { z } ) ) ~<_ { z } ) /\ ( y i^i { z } ) = (/) ) -> ( ( F " y ) u. ( _I ` ( F " { z } ) ) ) ~<_ ( y u. { z } ) )
57	43, 52, 55, 56	syl21anc 1222	. . . . . . . 8 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( ( F " y ) u. ( _I ` ( F " { z } ) ) ) ~<_ ( y u. { z } ) )
58	42, 57	eqbrtrrd 4462	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( F Fn A /\ ( F " y ) ~<_ y ) ) -> ( F " ( y u. { z } ) ) ~<_ ( y u. { z } ) )
59	58	exp32 605	. . . . . 6 |- ( ( y e. Fin /\ -. z e. y ) -> ( F Fn A -> ( ( F " y ) ~<_ y -> ( F " ( y u. { z } ) ) ~<_ ( y u. { z } ) ) ) )
60	59	a2d 26	. . . . 5 |- ( ( y e. Fin /\ -. z e. y ) -> ( ( F Fn A -> ( F " y ) ~<_ y ) -> ( F Fn A -> ( F " ( y u. { z } ) ) ~<_ ( y u. { z } ) ) ) )
61	9, 13, 17, 21, 24, 60	findcard2s 7750	. . . 4 |- ( A e. Fin -> ( F Fn A -> ( F " A ) ~<_ A ) )
62	3, 61	syl5 32	. . 3 |- ( A e. Fin -> ( F : A -onto-> B -> ( F " A ) ~<_ A ) )
63	62	imp 429	. 2 |- ( ( A e. Fin /\ F : A -onto-> B ) -> ( F " A ) ~<_ A )
64	2, 63	eqbrtrrd 4462	1 |- ( ( A e. Fin /\ F : A -onto-> B ) -> B ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   _Vcvv 3106   u. cun 3467   i^i cin 3468   C_ wss 3469   (/)c0 3778   {csn 4020   <.cop 4026   class class class wbr 4440   _I cid 4783   ran crn 4993   |` cres 4994   "cima 4995   Fun wfun 5573   Fn wfn 5574   -onto->wfo 5577   ` cfv 5579   1oc1o 7113   ~~ cen 7503   ~<_ cdom 7504   Fincfn 7506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-fin 7510

This theorem is referenced by:  fodomfib  7789  fofinf1o  7790  fidomdm  7791  fofi  7795  pwfilem  7803  cmpsub  19659  alexsubALT  20279