Theorem fopwdom 6681

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 4848	. . . . . 6 |- ( `' F " a ) C_ ran `' F
2		dfdm4 4689	. . . . . . 7 |- dom F = ran `' F
3		fof 5301	. . . . . . . 8 |- ( F : A -onto-> B -> F : A --> B )
4		fdm 5234	. . . . . . . 8 |- ( F : A --> B -> dom F = A )
5	3, 4	syl 14	. . . . . . 7 |- ( F : A -onto-> B -> dom F = A )
6	2, 5	syl5eqr 2159	. . . . . 6 |- ( F : A -onto-> B -> ran `' F = A )
7	1, 6	sseqtrid 3111	. . . . 5 |- ( F : A -onto-> B -> ( `' F " a ) C_ A )
8	7	adantl 273	. . . 4 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( `' F " a ) C_ A )
9		cnvexg 5032	. . . . . 6 |- ( F e. _V -> `' F e. _V )
10	9	adantr 272	. . . . 5 |- ( ( F e. _V /\ F : A -onto-> B ) -> `' F e. _V )
11		imaexg 4849	. . . . 5 |- ( `' F e. _V -> ( `' F " a ) e. _V )
12		elpwg 3482	. . . . 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 166	. . 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 4833	. . . . . . 7 |- ( ( `' F " a ) = ( `' F " b ) -> ( F " ( `' F " a ) ) = ( F " ( `' F " b ) ) )
17	16	adantl 273	. . . . . 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 506	. . . . . . 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 507	. . . . . . . 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 3485	. . . . . . 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 5339	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
22	18, 20, 21	syl2anc 406	. . . . . 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 508	. . . . . . . 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 3485	. . . . . . 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 5339	. . . . . . 7 |- ( ( F : A -onto-> B /\ b C_ B ) -> ( F " ( `' F " b ) ) = b )
26	18, 24, 25	syl2anc 406	. . . . . 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 2153	. . . . 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 114	. . . 4 |- ( ( ( F e. _V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) -> ( ( `' F " a ) = ( `' F " b ) -> a = b ) )
29		imaeq2 4833	. . . 4 |- ( a = b -> ( `' F " a ) = ( `' F " b ) )
30	28, 29	impbid1 141	. . 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 114	. 2 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( ( a e. ~P B /\ b e. ~P B ) -> ( ( `' F " a ) = ( `' F " b ) <-> a = b ) ) )
32		rnexg 4760	. . . . 5 |- ( F e. _V -> ran F e. _V )
33		forn 5304	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
34	33	eleq1d 2181	. . . . 5 |- ( F : A -onto-> B -> ( ran F e. _V <-> B e. _V ) )
35	32, 34	syl5ibcom 154	. . . 4 |- ( F e. _V -> ( F : A -onto-> B -> B e. _V ) )
36	35	imp 123	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> B e. _V )
37		pwexg 4062	. . 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 5268	. . . 4 |- ( ( F e. _V /\ F : A --> B ) -> A e. _V )
40	3, 39	sylan2 282	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> A e. _V )
41		pwexg 4062	. . 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 6620	1 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   _Vcvv 2655   C_ wss 3035   ~Pcpw 3474   class class class wbr 3893   `'ccnv 4496   dom cdm 4497   ran crn 4498   "cima 4500   -->wf 5075   -onto->wfo 5077   ~<_ cdom 6585

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-fv 5087  df-dom 6588

This theorem is referenced by: (None)