Theorem enprmaplem5 6081

Description: Lemma for enprmap 6083. Establish that ~PB is a subset of the range of W. (Contributed by SF, 3-Mar-2015.)

Hypotheses

Ref	Expression
enprmaplem5.1	|- W = (r e. (A ^m B) |-> (`'r"{x}))
enprmaplem5.2	|- R = (u e. B |-> if(u e. p, x, y))
enprmaplem5.3	|- B e. _V

Assertion

Ref	Expression
enprmaplem5	|- ((x =/= y /\ A = {x, y}) -> ~PB C_ ran W)

Distinct variable groups:   A,p   A,r   u,A   B,p   B,r   u,B,p   x,p   y,p   R,r   x,r   x,u   y,u   W,p

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

Proof of Theorem enprmaplem5

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 |- p e. _V
2	1	elpw 3729	. . 3 |- (p e. ~PB <-> p C_ B)
3		ifeqor 3700	. . . . . . . . . . . . . 14 |- (if(u e. p, x, y) = x \/ if(u e. p, x, y) = y)
4		vex 2863	. . . . . . . . . . . . . . . 16 |- x e. _V
5		vex 2863	. . . . . . . . . . . . . . . 16 |- y e. _V
6	4, 5	ifex 3721	. . . . . . . . . . . . . . 15 |- if(u e. p, x, y) e. _V
7	6	elpr 3752	. . . . . . . . . . . . . 14 |- (if(u e. p, x, y) e. {x, y} <-> (if(u e. p, x, y) = x \/ if(u e. p, x, y) = y))
8	3, 7	mpbir 200	. . . . . . . . . . . . 13 |- if(u e. p, x, y) e. {x, y}
9		id 19	. . . . . . . . . . . . 13 |- (A = {x, y} -> A = {x, y})
10	8, 9	syl5eleqr 2440	. . . . . . . . . . . 12 |- (A = {x, y} -> if(u e. p, x, y) e. A)
11	10	ralrimivw 2699	. . . . . . . . . . 11 |- (A = {x, y} -> A.u e. B if(u e. p, x, y) e. A)
12		enprmaplem5.2	. . . . . . . . . . . 12 |- R = (u e. B |-> if(u e. p, x, y))
13	12	fmpt 5693	. . . . . . . . . . 11 |- (A.u e. B if(u e. p, x, y) e. A <-> R:B-->A)
14	11, 13	sylib 188	. . . . . . . . . 10 |- (A = {x, y} -> R:B-->A)
15		prex 4113	. . . . . . . . . . . 12 |- {x, y} e. _V
16		eleq1 2413	. . . . . . . . . . . 12 |- (A = {x, y} -> (A e. _V <-> {x, y} e. _V))
17	15, 16	mpbiri 224	. . . . . . . . . . 11 |- (A = {x, y} -> A e. _V)
18		enprmaplem5.3	. . . . . . . . . . . 12 |- B e. _V
19	12, 18	enprmaplem4 6080	. . . . . . . . . . . 12 |- R e. _V
20		elmapg 6013	. . . . . . . . . . . 12 |- ((A e. _V /\ B e. _V /\ R e. _V) -> (R e. (A ^m B) <-> R:B-->A))
21	18, 19, 20	mp3an23 1269	. . . . . . . . . . 11 |- (A e. _V -> (R e. (A ^m B) <-> R:B-->A))
22	17, 21	syl 15	. . . . . . . . . 10 |- (A = {x, y} -> (R e. (A ^m B) <-> R:B-->A))
23	14, 22	mpbird 223	. . . . . . . . 9 |- (A = {x, y} -> R e. (A ^m B))
24	23	3ad2ant2 977	. . . . . . . 8 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> R e. (A ^m B))
25		cnveq 4887	. . . . . . . . . 10 |- (r = R -> `'r = `'R)
26	25	imaeq1d 4942	. . . . . . . . 9 |- (r = R -> (`'r"{x}) = (`'R"{x}))
27		enprmaplem5.1	. . . . . . . . 9 |- W = (r e. (A ^m B) |-> (`'r"{x}))
28	19	cnvex 5103	. . . . . . . . . 10 |- `'R e. _V
29		snex 4112	. . . . . . . . . 10 |- {x} e. _V
30	28, 29	imaex 4748	. . . . . . . . 9 |- (`'R"{x}) e. _V
31	26, 27, 30	fvmpt 5701	. . . . . . . 8 |- (R e. (A ^m B) -> (W` R) = (`'R"{x}))
32	24, 31	syl 15	. . . . . . 7 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (W` R) = (`'R"{x}))
33		eliniseg 5021	. . . . . . . . 9 |- (z e. (`'R"{x}) <-> zRx)
34		breldm 4912	. . . . . . . . . . . . 13 |- (zRx -> z e. dom R)
35	12	fnmpt 5690	. . . . . . . . . . . . . . 15 |- (A.u e. B if(u e. p, x, y) e. _V -> R Fn B)
36	6	a1i 10	. . . . . . . . . . . . . . 15 |- (u e. B -> if(u e. p, x, y) e. _V)
37	35, 36	mprg 2684	. . . . . . . . . . . . . 14 |- R Fn B
38		fndm 5183	. . . . . . . . . . . . . 14 |- (R Fn B -> dom R = B)
39	37, 38	ax-mp 5	. . . . . . . . . . . . 13 |- dom R = B
40	34, 39	syl6eleq 2443	. . . . . . . . . . . 12 |- (zRx -> z e. B)
41		fnbrfvb 5359	. . . . . . . . . . . . . . 15 |- ((R Fn B /\ z e. B) -> ((R` z) = x <-> zRx))
42	37, 41	mpan 651	. . . . . . . . . . . . . 14 |- (z e. B -> ((R` z) = x <-> zRx))
43	42	biimprd 214	. . . . . . . . . . . . 13 |- (z e. B -> (zRx -> (R` z) = x))
44	43	com12 27	. . . . . . . . . . . 12 |- (zRx -> (z e. B -> (R` z) = x))
45	40, 44	jcai 522	. . . . . . . . . . 11 |- (zRx -> (z e. B /\ (R` z) = x))
46		eleq1 2413	. . . . . . . . . . . . . . . . 17 |- (u = z -> (u e. p <-> z e. p))
47	46	ifbid 3681	. . . . . . . . . . . . . . . 16 |- (u = z -> if(u e. p, x, y) = if(z e. p, x, y))
48	4, 5	ifex 3721	. . . . . . . . . . . . . . . 16 |- if(z e. p, x, y) e. _V
49	47, 12, 48	fvmpt 5701	. . . . . . . . . . . . . . 15 |- (z e. B -> (R` z) = if(z e. p, x, y))
50	49	eqeq1d 2361	. . . . . . . . . . . . . 14 |- (z e. B -> ((R` z) = x <-> if(z e. p, x, y) = x))
51	50	biimpd 198	. . . . . . . . . . . . 13 |- (z e. B -> ((R` z) = x -> if(z e. p, x, y) = x))
52	51	imp 418	. . . . . . . . . . . 12 |- ((z e. B /\ (R` z) = x) -> if(z e. p, x, y) = x)
53		simpl1 958	. . . . . . . . . . . . . . 15 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ if(z e. p, x, y) = x) -> x =/= y)
54		df-ne 2519	. . . . . . . . . . . . . . 15 |- (x =/= y <-> -. x = y)
55	53, 54	sylib 188	. . . . . . . . . . . . . 14 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ if(z e. p, x, y) = x) -> -. x = y)
56		iffalse 3670	. . . . . . . . . . . . . . . . . . 19 |- (-. z e. p -> if(z e. p, x, y) = y)
57	56	eqeq2d 2364	. . . . . . . . . . . . . . . . . 18 |- (-. z e. p -> (x = if(z e. p, x, y) <-> x = y))
58	57	biimpd 198	. . . . . . . . . . . . . . . . 17 |- (-. z e. p -> (x = if(z e. p, x, y) -> x = y))
59	58	com12 27	. . . . . . . . . . . . . . . 16 |- (x = if(z e. p, x, y) -> (-. z e. p -> x = y))
60	59	eqcoms 2356	. . . . . . . . . . . . . . 15 |- (if(z e. p, x, y) = x -> (-. z e. p -> x = y))
61	60	adantl 452	. . . . . . . . . . . . . 14 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ if(z e. p, x, y) = x) -> (-. z e. p -> x = y))
62	55, 61	mt3d 117	. . . . . . . . . . . . 13 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ if(z e. p, x, y) = x) -> z e. p)
63	62	ex 423	. . . . . . . . . . . 12 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (if(z e. p, x, y) = x -> z e. p))
64	52, 63	syl5 28	. . . . . . . . . . 11 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> ((z e. B /\ (R` z) = x) -> z e. p))
65	45, 64	syl5 28	. . . . . . . . . 10 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (zRx -> z e. p))
66		ssel2 3269	. . . . . . . . . . . . . . 15 |- ((p C_ B /\ z e. p) -> z e. B)
67	66	3ad2antl3 1119	. . . . . . . . . . . . . 14 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> z e. B)
68	67, 49	syl 15	. . . . . . . . . . . . 13 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> (R` z) = if(z e. p, x, y))
69		iftrue 3669	. . . . . . . . . . . . . 14 |- (z e. p -> if(z e. p, x, y) = x)
70	69	adantl 452	. . . . . . . . . . . . 13 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> if(z e. p, x, y) = x)
71	68, 70	eqtrd 2385	. . . . . . . . . . . 12 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> (R` z) = x)
72	67, 42	syl 15	. . . . . . . . . . . 12 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> ((R` z) = x <-> zRx))
73	71, 72	mpbid 201	. . . . . . . . . . 11 |- (((x =/= y /\ A = {x, y} /\ p C_ B) /\ z e. p) -> zRx)
74	73	ex 423	. . . . . . . . . 10 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (z e. p -> zRx))
75	65, 74	impbid 183	. . . . . . . . 9 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (zRx <-> z e. p))
76	33, 75	syl5bb 248	. . . . . . . 8 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (z e. (`'R"{x}) <-> z e. p))
77	76	eqrdv 2351	. . . . . . 7 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (`'R"{x}) = p)
78	32, 77	eqtrd 2385	. . . . . 6 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> (W` R) = p)
79	27	enprmaplem2 6078	. . . . . . . 8 |- W Fn (A ^m B)
80		fnbrfvb 5359	. . . . . . . 8 |- ((W Fn (A ^m B) /\ R e. (A ^m B)) -> ((W` R) = p <-> RWp))
81	79, 80	mpan 651	. . . . . . 7 |- (R e. (A ^m B) -> ((W` R) = p <-> RWp))
82	24, 81	syl 15	. . . . . 6 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> ((W` R) = p <-> RWp))
83	78, 82	mpbid 201	. . . . 5 |- ((x =/= y /\ A = {x, y} /\ p C_ B) -> RWp)
84	83	3expia 1153	. . . 4 |- ((x =/= y /\ A = {x, y}) -> (p C_ B -> RWp))
85		brelrn 4961	. . . 4 |- (RWp -> p e. ran W)
86	84, 85	syl6 29	. . 3 |- ((x =/= y /\ A = {x, y}) -> (p C_ B -> p e. ran W))
87	2, 86	syl5bi 208	. 2 |- ((x =/= y /\ A = {x, y}) -> (p e. ~PB -> p e. ran W))
88	87	ssrdv 3279	1 |- ((x =/= y /\ A = {x, y}) -> ~PB C_ ran W)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  _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:  enprmaplem6  6082