Theorem fopwdom 6246

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) -> ~PB ~<_ ~PA)

Proof of Theorem fopwdom

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

Step	Hyp	Ref	Expression
1		imassrn 4622	. . . . . 6 |- ( `' F " a) C_ ran `' F
2		dfdm4 4470	. . . . . . 7 |- dom F = ran `' F
3		fof 5049	. . . . . . . 8 |- ( F :A -onto->B -> F :A -->B)
4		fdm 4993	. . . . . . . 8 |- ( F :A -->B -> dom F = A)
5	3, 4	syl 14	. . . . . . 7 |- ( F :A -onto->B -> dom F = A)
6	2, 5	syl5eqr 2083	. . . . . 6 |- ( F :A -onto->B -> ran `' F = A)
7	1, 6	syl5sseq 2987	. . . . 5 |- ( F :A -onto->B -> ( `' F " a) C_ A)
8	7	adantl 262	. . . 4 |- (( F e. _V /\ F :A -onto->B) -> ( `' F " a) C_ A)
9		cnvexg 4798	. . . . . 6 |- ( F e. _V -> `' F e. _V)
10	9	adantr 261	. . . . 5 |- (( F e. _V /\ F :A -onto->B) -> `' F e. _V)
11		imaexg 4623	. . . . 5 |- ( `' F e. _V -> ( `' F " a) e. _V)
12		elpwg 3359	. . . . 5 |- (( `' F " a) e. _V -> (( `' F " a) e. ~PA <-> ( `' F " a) C_ A))
13	10, 11, 12	3syl 17	. . . 4 |- (( F e. _V /\ F :A -onto->B) -> (( `' F " a) e. ~PA <-> ( `' F " a) C_ A))
14	8, 13	mpbird 156	. . 3 |- (( F e. _V /\ F :A -onto->B) -> ( `' F " a) e. ~PA)
15	14	a1d 22	. 2 |- (( F e. _V /\ F :A -onto->B) -> ( a e. ~PB -> ( `' F " a) e. ~PA))
16		imaeq2 4607	. . . . . . 7 |- (( `' F " a) = ( `' F " b) -> ( F "( `' F " a)) = ( F "( `' F " b)))
17	16	adantl 262	. . . . . 6 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> ( F "( `' F " a)) = ( F "( `' F " b)))
18		simpllr 486	. . . . . . 7 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> F :A -onto->B)
19		simplrl 487	. . . . . . . 8 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> a e. ~PB)
20	19	elpwid 3361	. . . . . . 7 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> a C_ B)
21		foimacnv 5087	. . . . . . 7 |- (( F :A -onto->B /\ a C_ B) -> ( F "( `' F " a)) = a)
22	18, 20, 21	syl2anc 391	. . . . . 6 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> ( F "( `' F " a)) = a)
23		simplrr 488	. . . . . . . 8 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> b e. ~PB)
24	23	elpwid 3361	. . . . . . 7 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> b C_ B)
25		foimacnv 5087	. . . . . . 7 |- (( F :A -onto->B /\ b C_ B) -> ( F "( `' F " b)) = b)
26	18, 24, 25	syl2anc 391	. . . . . 6 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> ( F "( `' F " b)) = b)
27	17, 22, 26	3eqtr3d 2077	. . . . 5 |- (((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) /\ ( `' F " a) = ( `' F " b)) -> a = b)
28	27	ex 108	. . . 4 |- ((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) -> (( `' F " a) = ( `' F " b) -> a = b))
29		imaeq2 4607	. . . 4 |- ( a = b -> ( `' F " a) = ( `' F " b))
30	28, 29	impbid1 130	. . 3 |- ((( F e. _V /\ F :A -onto->B) /\ ( a e. ~PB /\ b e. ~PB)) -> (( `' F " a) = ( `' F " b) <-> a = b))
31	30	ex 108	. 2 |- (( F e. _V /\ F :A -onto->B) -> (( a e. ~PB /\ b e. ~PB) -> (( `' F " a) = ( `' F " b) <-> a = b)))
32		rnexg 4540	. . . . 5 |- ( F e. _V -> ran F e. _V)
33		forn 5052	. . . . . 6 |- ( F :A -onto->B -> ran F = B)
34	33	eleq1d 2103	. . . . 5 |- ( F :A -onto->B -> ( ran F e. _V <-> B e. _V))
35	32, 34	syl5ibcom 144	. . . 4 |- ( F e. _V -> ( F :A -onto->B -> B e. _V))
36	35	imp 115	. . 3 |- (( F e. _V /\ F :A -onto->B) -> B e. _V)
37		pwexg 3924	. . 3 |- (B e. _V -> ~PB e. _V)
38	36, 37	syl 14	. 2 |- (( F e. _V /\ F :A -onto->B) -> ~PB e. _V)
39		dmfex 5022	. . . 4 |- (( F e. _V /\ F :A -->B) -> A e. _V)
40	3, 39	sylan2 270	. . 3 |- (( F e. _V /\ F :A -onto->B) -> A e. _V)
41		pwexg 3924	. . 3 |- (A e. _V -> ~PA e. _V)
42	40, 41	syl 14	. 2 |- (( F e. _V /\ F :A -onto->B) -> ~PA e. _V)
43	15, 31, 38, 42	dom3d 6190	1 |- (( F e. _V /\ F :A -onto->B) -> ~PB ~<_ ~PA)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   _Vcvv 2551   C_ wss 2911   ~Pcpw 3351   class class class wbr 3755   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291   -->wf 4841   -onto->wfo 4843   ~<_ cdom 6156

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-fv 4853  df-dom 6159

This theorem is referenced by: (None)