Theorem fopwdom 8068

Description: Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.)

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 5477	. . . . . 6 |- ( `' F " a ) C_ ran `' F
2		dfdm4 5316	. . . . . . 7 |- dom F = ran `' F
3		fof 6115	. . . . . . . 8 |- ( F : A -onto-> B -> F : A --> B )
4		fdm 6051	. . . . . . . 8 |- ( F : A --> B -> dom F = A )
5	3, 4	syl 17	. . . . . . 7 |- ( F : A -onto-> B -> dom F = A )
6	2, 5	syl5eqr 2670	. . . . . 6 |- ( F : A -onto-> B -> ran `' F = A )
7	1, 6	syl5sseq 3653	. . . . 5 |- ( F : A -onto-> B -> ( `' F " a ) C_ A )
8	7	adantl 482	. . . 4 |- ( ( F e. V /\ F : A -onto-> B ) -> ( `' F " a ) C_ A )
9		cnvexg 7112	. . . . . 6 |- ( F e. V -> `' F e. _V )
10	9	adantr 481	. . . . 5 |- ( ( F e. V /\ F : A -onto-> B ) -> `' F e. _V )
11		imaexg 7103	. . . . 5 |- ( `' F e. _V -> ( `' F " a ) e. _V )
12		elpwg 4166	. . . . 5 |- ( ( `' F " a ) e. _V -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
13	10, 11, 12	3syl 18	. . . 4 |- ( ( F e. V /\ F : A -onto-> B ) -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
14	8, 13	mpbird 247	. . 3 |- ( ( F e. V /\ F : A -onto-> B ) -> ( `' F " a ) e. ~P A )
15	14	a1d 25	. 2 |- ( ( F e. V /\ F : A -onto-> B ) -> ( a e. ~P B -> ( `' F " a ) e. ~P A ) )
16		imaeq2 5462	. . . . . . 7 |- ( ( `' F " a ) = ( `' F " b ) -> ( F " ( `' F " a ) ) = ( F " ( `' F " b ) ) )
17	16	adantl 482	. . . . . 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 799	. . . . . . 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 800	. . . . . . . 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 4170	. . . . . . 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 6154	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
22	18, 20, 21	syl2anc 693	. . . . . 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 801	. . . . . . . 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 4170	. . . . . . 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 6154	. . . . . . 7 |- ( ( F : A -onto-> B /\ b C_ B ) -> ( F " ( `' F " b ) ) = b )
26	18, 24, 25	syl2anc 693	. . . . . 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 2664	. . . . 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 450	. . . 4 |- ( ( ( F e. V /\ F : A -onto-> B ) /\ ( a e. ~P B /\ b e. ~P B ) ) -> ( ( `' F " a ) = ( `' F " b ) -> a = b ) )
29		imaeq2 5462	. . . 4 |- ( a = b -> ( `' F " a ) = ( `' F " b ) )
30	28, 29	impbid1 215	. . 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 450	. 2 |- ( ( F e. V /\ F : A -onto-> B ) -> ( ( a e. ~P B /\ b e. ~P B ) -> ( ( `' F " a ) = ( `' F " b ) <-> a = b ) ) )
32		rnexg 7098	. . . . 5 |- ( F e. V -> ran F e. _V )
33		forn 6118	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
34	33	eleq1d 2686	. . . . 5 |- ( F : A -onto-> B -> ( ran F e. _V <-> B e. _V ) )
35	32, 34	syl5ibcom 235	. . . 4 |- ( F e. V -> ( F : A -onto-> B -> B e. _V ) )
36	35	imp 445	. . 3 |- ( ( F e. V /\ F : A -onto-> B ) -> B e. _V )
37		pwexg 4850	. . 3 |- ( B e. _V -> ~P B e. _V )
38	36, 37	syl 17	. 2 |- ( ( F e. V /\ F : A -onto-> B ) -> ~P B e. _V )
39		dmfex 7124	. . . 4 |- ( ( F e. V /\ F : A --> B ) -> A e. _V )
40	3, 39	sylan2 491	. . 3 |- ( ( F e. V /\ F : A -onto-> B ) -> A e. _V )
41		pwexg 4850	. . 3 |- ( A e. _V -> ~P A e. _V )
42	40, 41	syl 17	. 2 |- ( ( F e. V /\ F : A -onto-> B ) -> ~P A e. _V )
43	15, 31, 38, 42	dom3d 7997	1 |- ( ( F e. V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   -onto->wfo 5886   ~<_ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-fv 5896  df-dom 7957

This theorem is referenced by:  pwdom  8112  wdompwdom  8483