Theorem enprmaplem6 6082

Description: Lemma for enprmap 6083. The range of W is ~PB. (Contributed by SF, 3-Mar-2015.)

Hypotheses

Ref	Expression
enprmaplem6.1	|- W = (r e. (A ^m B) |-> (`'r"{x}))
enprmaplem6.2	|- B e. _V

Assertion

Ref	Expression
enprmaplem6	|- ((x =/= y /\ A = {x, y}) -> ran W = ~PB)

Distinct variable groups:   A,r   B,r   x,r   y,r

Allowed substitution hints:   A(x,y)   B(x,y)   W(x,y,r)

Proof of Theorem enprmaplem6

Dummy variables p s u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breldm 4912	. . . . . . . 8 |- (sWp -> s e. dom W)
2		enprmaplem6.1	. . . . . . . . . 10 |- W = (r e. (A ^m B) |-> (`'r"{x}))
3	2	enprmaplem2 6078	. . . . . . . . 9 |- W Fn (A ^m B)
4		fndm 5183	. . . . . . . . 9 |- (W Fn (A ^m B) -> dom W = (A ^m B))
5	3, 4	ax-mp 5	. . . . . . . 8 |- dom W = (A ^m B)
6	1, 5	syl6eleq 2443	. . . . . . 7 |- (sWp -> s e. (A ^m B))
7		fnbrfvb 5359	. . . . . . . . 9 |- ((W Fn (A ^m B) /\ s e. (A ^m B)) -> ((W` s) = p <-> sWp))
8	3, 6, 7	sylancr 644	. . . . . . . 8 |- (sWp -> ((W` s) = p <-> sWp))
9	8	ibir 233	. . . . . . 7 |- (sWp -> (W` s) = p)
10	6, 9	jca 518	. . . . . 6 |- (sWp -> (s e. (A ^m B) /\ (W` s) = p))
11		cnveq 4887	. . . . . . . . . . . . 13 |- (r = s -> `'r = `'s)
12	11	imaeq1d 4942	. . . . . . . . . . . 12 |- (r = s -> (`'r"{x}) = (`'s"{x}))
13		vex 2863	. . . . . . . . . . . . . 14 |- s e. _V
14	13	cnvex 5103	. . . . . . . . . . . . 13 |- `'s e. _V
15		snex 4112	. . . . . . . . . . . . 13 |- {x} e. _V
16	14, 15	imaex 4748	. . . . . . . . . . . 12 |- (`'s"{x}) e. _V
17	12, 2, 16	fvmpt 5701	. . . . . . . . . . 11 |- (s e. (A ^m B) -> (W` s) = (`'s"{x}))
18	17	eqeq1d 2361	. . . . . . . . . 10 |- (s e. (A ^m B) -> ((W` s) = p <-> (`'s"{x}) = p))
19	18	3ad2ant3 978	. . . . . . . . 9 |- ((x =/= y /\ A = {x, y} /\ s e. (A ^m B)) -> ((W` s) = p <-> (`'s"{x}) = p))
20		imassrn 5010	. . . . . . . . . . . 12 |- (`'s"{x}) C_ ran `'s
21		df-dm 4788	. . . . . . . . . . . . 13 |- dom s = ran `'s
22		elmapi 6017	. . . . . . . . . . . . . 14 |- (s e. (A ^m B) -> s:B-->A)
23		fdm 5227	. . . . . . . . . . . . . 14 |- (s:B-->A -> dom s = B)
24		eqimss 3324	. . . . . . . . . . . . . 14 |- (dom s = B -> dom s C_ B)
25	22, 23, 24	3syl 18	. . . . . . . . . . . . 13 |- (s e. (A ^m B) -> dom s C_ B)
26	21, 25	syl5eqssr 3317	. . . . . . . . . . . 12 |- (s e. (A ^m B) -> ran `'s C_ B)
27	20, 26	syl5ss 3284	. . . . . . . . . . 11 |- (s e. (A ^m B) -> (`'s"{x}) C_ B)
28	27	3ad2ant3 978	. . . . . . . . . 10 |- ((x =/= y /\ A = {x, y} /\ s e. (A ^m B)) -> (`'s"{x}) C_ B)
29		sseq1 3293	. . . . . . . . . 10 |- ((`'s"{x}) = p -> ((`'s"{x}) C_ B <-> p C_ B))
30	28, 29	syl5ibcom 211	. . . . . . . . 9 |- ((x =/= y /\ A = {x, y} /\ s e. (A ^m B)) -> ((`'s"{x}) = p -> p C_ B))
31	19, 30	sylbid 206	. . . . . . . 8 |- ((x =/= y /\ A = {x, y} /\ s e. (A ^m B)) -> ((W` s) = p -> p C_ B))
32	31	3expia 1153	. . . . . . 7 |- ((x =/= y /\ A = {x, y}) -> (s e. (A ^m B) -> ((W` s) = p -> p C_ B)))
33	32	imp3a 420	. . . . . 6 |- ((x =/= y /\ A = {x, y}) -> ((s e. (A ^m B) /\ (W` s) = p) -> p C_ B))
34	10, 33	syl5 28	. . . . 5 |- ((x =/= y /\ A = {x, y}) -> (sWp -> p C_ B))
35	34	exlimdv 1636	. . . 4 |- ((x =/= y /\ A = {x, y}) -> (E.s sWp -> p C_ B))
36		elrn 4897	. . . 4 |- (p e. ran W <-> E.s sWp)
37		vex 2863	. . . . 5 |- p e. _V
38	37	elpw 3729	. . . 4 |- (p e. ~PB <-> p C_ B)
39	35, 36, 38	3imtr4g 261	. . 3 |- ((x =/= y /\ A = {x, y}) -> (p e. ran W -> p e. ~PB))
40	39	ssrdv 3279	. 2 |- ((x =/= y /\ A = {x, y}) -> ran W C_ ~PB)
41		eqid 2353	. . 3 |- (u e. B |-> if(u e. p, x, y)) = (u e. B |-> if(u e. p, x, y))
42		enprmaplem6.2	. . 3 |- B e. _V
43	2, 41, 42	enprmaplem5 6081	. 2 |- ((x =/= y /\ A = {x, y}) -> ~PB C_ ran W)
44	40, 43	eqssd 3290	1 |- ((x =/= y /\ A = {x, y}) -> ran W = ~PB)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860   C_ wss 3258  ifcif 3663  ~Pcpw 3723  {csn 3738  {cpr 3739   class class class wbr 4640  "cima 4723  `'ccnv 4772  dom cdm 4773  ran crn 4774   Fn wfn 4777  -->wf 4778  ` cfv 4782  (class class class)co 5526   |-> cmpt 5652   ^m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by:  enprmap  6083