Theorem fopwdom 6723

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 4887	. . . . . 6 |- ( `' F " a ) C_ ran `' F
2		dfdm4 4726	. . . . . . 7 |- dom F = ran `' F
3		fof 5340	. . . . . . . 8 |- ( F : A -onto-> B -> F : A --> B )
4		fdm 5273	. . . . . . . 8 |- ( F : A --> B -> dom F = A )
5	3, 4	syl 14	. . . . . . 7 |- ( F : A -onto-> B -> dom F = A )
6	2, 5	syl5eqr 2184	. . . . . 6 |- ( F : A -onto-> B -> ran `' F = A )
7	1, 6	sseqtrid 3142	. . . . 5 |- ( F : A -onto-> B -> ( `' F " a ) C_ A )
8	7	adantl 275	. . . 4 |- ( ( F e. _V /\ F : A -onto-> B ) -> ( `' F " a ) C_ A )
9		cnvexg 5071	. . . . . 6 |- ( F e. _V -> `' F e. _V )
10	9	adantr 274	. . . . 5 |- ( ( F e. _V /\ F : A -onto-> B ) -> `' F e. _V )
11		imaexg 4888	. . . . 5 |- ( `' F e. _V -> ( `' F " a ) e. _V )
12		elpwg 3513	. . . . 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 4872	. . . . . . 7 |- ( ( `' F " a ) = ( `' F " b ) -> ( F " ( `' F " a ) ) = ( F " ( `' F " b ) ) )
17	16	adantl 275	. . . . . 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 523	. . . . . . 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 524	. . . . . . . 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 3516	. . . . . . 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 5378	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
22	18, 20, 21	syl2anc 408	. . . . . 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 525	. . . . . . . 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 3516	. . . . . . 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 5378	. . . . . . 7 |- ( ( F : A -onto-> B /\ b C_ B ) -> ( F " ( `' F " b ) ) = b )
26	18, 24, 25	syl2anc 408	. . . . . 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 2178	. . . . 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 4872	. . . 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 4799	. . . . 5 |- ( F e. _V -> ran F e. _V )
33		forn 5343	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
34	33	eleq1d 2206	. . . . 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 4099	. . 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 5307	. . . 4 |- ( ( F e. _V /\ F : A --> B ) -> A e. _V )
40	3, 39	sylan2 284	. . 3 |- ( ( F e. _V /\ F : A -onto-> B ) -> A e. _V )
41		pwexg 4099	. . 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 6661	1 |- ( ( F e. _V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2681   C_ wss 3066   ~Pcpw 3505   class class class wbr 3924   `'ccnv 4533   dom cdm 4534   ran crn 4535   "cima 4537   -->wf 5114   -onto->wfo 5116   ~<_ cdom 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-fv 5126  df-dom 6629

This theorem is referenced by: (None)