Theorem fopwdom 6894

Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
fopwdom	|- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Proof of Theorem fopwdom

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5017	. . . . . 6 |- ( `' F " a ) C_ ran `' F
2		dfdm4 4855	. . . . . . 7 |- dom F = ran `' F
3		fof 5477	. . . . . . . 8 |- ( F : A -onto-> B -> F : A --> B )
4		fdm 5410	. . . . . . . 8 |- ( F : A --> B -> dom F = A )
5	3, 4	syl 14	. . . . . . 7 |- ( F : A -onto-> B -> dom F = A )
6	2, 5	eqtr3id 2240	. . . . . 6 |- ( F : A -onto-> B -> ran `' F = A )
7	1, 6	sseqtrid 3230	. . . . 5 |- ( F : A -onto-> B -> ( `' F " a ) C_ A )
8	7	adantl 277	. . . 4 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( `' F " a ) C_ A )
9		cnvexg 5204	. . . . . 6 |- ( F e. _V -> `' F e. _V )
10	9	adantr 276	. . . . 5 |- ( ( F e. _V /\ F : A -onto-> B ) -> `' F e. _V )
11		imaexg 5020	. . . . 5 |- ( `' F e. _V -> ( `' F " a ) e. _V )
12		elpwg 3610	. . . . 5 |- ( ( `' F " a ) e. _V -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
13	10, 11, 12	3syl 17	. . . 4 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
14	8, 13	mpbird 167	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( `' F " a ) e. ~P A )
15	14	a1d 22	. 2 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( a e. ~P B -> ( `' F " a ) e. ~P A ) )
16		imaeq2 5002	. . . . . . 7 |- ( ( `' F " a ) = ( `' F " b ) -> ( F " ( `' F " a ) ) = ( F " ( `' F " b ) ) )
17	16	adantl 277	. . . . . 6 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> ( F " ( `' F " a ) ) = ( F " ( `' F " b ) ) )
18		simpllr 534	. . . . . . 7 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> F : A -onto-> B )
19		simplrl 535	. . . . . . . 8 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> a e. ~P B )
20	19	elpwid 3613	. . . . . . 7 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> a C_ B )
21		foimacnv 5519	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
22	18, 20, 21	syl2anc 411	. . . . . 6 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> ( F " ( `' F " a ) ) = a )
23		simplrr 536	. . . . . . . 8 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> b e. ~P B )
24	23	elpwid 3613	. . . . . . 7 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> b C_ B )
25		foimacnv 5519	. . . . . . 7 |- ( ( F : A -onto-> B /\ b C_ B ) -> ( F " ( `' F " b ) ) = b )
26	18, 24, 25	syl2anc 411	. . . . . 6 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> ( F " ( `' F " b ) ) = b )
27	17, 22, 26	3eqtr3d 2234	. . . . 5 |- ( ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) /\ ( `' F " a ) = ( `' F " b ) ) -> a = b )
28	27	ex 115	. . . 4 |- ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) -> ( ( `' F " a ) = ( `' F " b ) -> a = b ) )
29		imaeq2 5002	. . . 4 |- ( a = b -> ( `' F " a ) = ( `' F " b ) )
30	28, 29	impbid1 142	. . 3 |- ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) -> ( ( `' F " a ) = ( `' F " b ) <-> a = b ) )
31	30	ex 115	. 2 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( ( a e. ~P B /\ b e. ~P B ) -> ( ( `' F " a ) = ( `' F " b ) <-> a = b ) ) )
32		rnexg 4928	. . . . 5 |- ( F e. _V -> ran F e. _V )
33		forn 5480	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
34	33	eleq1d 2262	. . . . 5 |- ( F : A -onto-> B -> ( ran F e. _V <-> B e. _V ) )
35	32, 34	syl5ibcom 155	. . . 4 |- ( F e. _V -> ( F : A -onto-> B -> B e. _V ) )
36	35	imp 124	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> B e. _V )
37		pwexg 4210	. . 3 |- ( B e. _V -> ~P B e. _V )
38	36, 37	syl 14	. 2 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B e. _V )
39		dmfex 5444	. . . 4 |- ( ( F e. _V /\ F : A --> B ) -> A e. _V )
40	3, 39	sylan2 286	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> A e. _V )
41		pwexg 4210	. . 3 |- ( A e. _V -> ~P A e. _V )
42	40, 41	syl 14	. 2 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P A e. _V )
43	15, 31, 38, 42	dom3d 6830	1 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   class class class wbr 4030   `'ccnv 4659   dom cdm 4660   ran crn 4661   "cima 4663   -->wf 5251   -onto->wfo 5253   ~<_ cdom 6795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-fv 5263  df-dom 6798

This theorem is referenced by: (None)