Theorem fopwdom 6838

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 4983	. . . . . 6 |- ( `' F " a ) C_ ran `' F
2		dfdm4 4821	. . . . . . 7 |- dom F = ran `' F
3		fof 5440	. . . . . . . 8 |- ( F : A -onto-> B -> F : A --> B )
4		fdm 5373	. . . . . . . 8 |- ( F : A --> B -> dom F = A )
5	3, 4	syl 14	. . . . . . 7 |- ( F : A -onto-> B -> dom F = A )
6	2, 5	eqtr3id 2224	. . . . . 6 |- ( F : A -onto-> B -> ran `' F = A )
7	1, 6	sseqtrid 3207	. . . . 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 5168	. . . . . 6 |- ( F e. _V -> `' F e. _V )
10	9	adantr 276	. . . . 5 |- ( ( F e. _V /\ F : A -onto-> B ) -> `' F e. _V )
11		imaexg 4984	. . . . 5 |- ( `' F e. _V -> ( `' F " a ) e. _V )
12		elpwg 3585	. . . . 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 4968	. . . . . . 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 3588	. . . . . . 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 5481	. . . . . . 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 3588	. . . . . . 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 5481	. . . . . . 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 2218	. . . . 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 4968	. . . 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 4894	. . . . 5 |- ( F e. _V -> ran F e. _V )
33		forn 5443	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
34	33	eleq1d 2246	. . . . 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 4182	. . 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 5407	. . . 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 4182	. . 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 6776	1 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   class class class wbr 4005   `'ccnv 4627   dom cdm 4628   ran crn 4629   "cima 4631   -->wf 5214   -onto->wfo 5216   ~<_ cdom 6741

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-fv 5226  df-dom 6744

This theorem is referenced by: (None)