Theorem enprmaplem6 6082

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

Hypotheses

Ref	Expression
enprmaplem6.1	⊢ W = (r ∈ (A ↑m B) ↦ (◡r “ {x}))
enprmaplem6.2	⊢ B ∈ V

Assertion

Ref	Expression
enprmaplem6	⊢ ((x ≠ y ∧ A = {x, y}) → ran W = ℘B)

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 ∈ dom W)
2		enprmaplem6.1	. . . . . . . . . 10 ⊢ W = (r ∈ (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 ∈ (A ↑m B))
7		fnbrfvb 5359	. . . . . . . . 9 ⊢ ((W Fn (A ↑m B) ∧ s ∈ (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 ∈ (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 ∈ V
14	13	cnvex 5103	. . . . . . . . . . . . 13 ⊢ ◡s ∈ V
15		snex 4112	. . . . . . . . . . . . 13 ⊢ {x} ∈ V
16	14, 15	imaex 4748	. . . . . . . . . . . 12 ⊢ (◡s “ {x}) ∈ V
17	12, 2, 16	fvmpt 5701	. . . . . . . . . . 11 ⊢ (s ∈ (A ↑m B) → (W ‘s) = (◡s “ {x}))
18	17	eqeq1d 2361	. . . . . . . . . 10 ⊢ (s ∈ (A ↑m B) → ((W ‘s) = p ↔ (◡s “ {x}) = p))
19	18	3ad2ant3 978	. . . . . . . . 9 ⊢ ((x ≠ y ∧ A = {x, y} ∧ s ∈ (A ↑m B)) → ((W ‘s) = p ↔ (◡s “ {x}) = p))
20		imassrn 5010	. . . . . . . . . . . 12 ⊢ (◡s “ {x}) ⊆ ran ◡s
21		df-dm 4788	. . . . . . . . . . . . 13 ⊢ dom s = ran ◡s
22		elmapi 6017	. . . . . . . . . . . . . 14 ⊢ (s ∈ (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 ⊆ B)
25	22, 23, 24	3syl 18	. . . . . . . . . . . . 13 ⊢ (s ∈ (A ↑m B) → dom s ⊆ B)
26	21, 25	syl5eqssr 3317	. . . . . . . . . . . 12 ⊢ (s ∈ (A ↑m B) → ran ◡s ⊆ B)
27	20, 26	syl5ss 3284	. . . . . . . . . . 11 ⊢ (s ∈ (A ↑m B) → (◡s “ {x}) ⊆ B)
28	27	3ad2ant3 978	. . . . . . . . . 10 ⊢ ((x ≠ y ∧ A = {x, y} ∧ s ∈ (A ↑m B)) → (◡s “ {x}) ⊆ B)
29		sseq1 3293	. . . . . . . . . 10 ⊢ ((◡s “ {x}) = p → ((◡s “ {x}) ⊆ B ↔ p ⊆ B))
30	28, 29	syl5ibcom 211	. . . . . . . . 9 ⊢ ((x ≠ y ∧ A = {x, y} ∧ s ∈ (A ↑m B)) → ((◡s “ {x}) = p → p ⊆ B))
31	19, 30	sylbid 206	. . . . . . . 8 ⊢ ((x ≠ y ∧ A = {x, y} ∧ s ∈ (A ↑m B)) → ((W ‘s) = p → p ⊆ B))
32	31	3expia 1153	. . . . . . 7 ⊢ ((x ≠ y ∧ A = {x, y}) → (s ∈ (A ↑m B) → ((W ‘s) = p → p ⊆ B)))
33	32	imp3a 420	. . . . . 6 ⊢ ((x ≠ y ∧ A = {x, y}) → ((s ∈ (A ↑m B) ∧ (W ‘s) = p) → p ⊆ B))
34	10, 33	syl5 28	. . . . 5 ⊢ ((x ≠ y ∧ A = {x, y}) → (sWp → p ⊆ B))
35	34	exlimdv 1636	. . . 4 ⊢ ((x ≠ y ∧ A = {x, y}) → (∃s sWp → p ⊆ B))
36		elrn 4897	. . . 4 ⊢ (p ∈ ran W ↔ ∃s sWp)
37		vex 2863	. . . . 5 ⊢ p ∈ V
38	37	elpw 3729	. . . 4 ⊢ (p ∈ ℘B ↔ p ⊆ B)
39	35, 36, 38	3imtr4g 261	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → (p ∈ ran W → p ∈ ℘B))
40	39	ssrdv 3279	. 2 ⊢ ((x ≠ y ∧ A = {x, y}) → ran W ⊆ ℘B)
41		eqid 2353	. . 3 ⊢ (u ∈ B ↦ if(u ∈ p, x, y)) = (u ∈ B ↦ if(u ∈ p, x, y))
42		enprmaplem6.2	. . 3 ⊢ B ∈ V
43	2, 41, 42	enprmaplem5 6081	. 2 ⊢ ((x ≠ y ∧ A = {x, y}) → ℘B ⊆ ran W)
44	40, 43	eqssd 3290	1 ⊢ ((x ≠ y ∧ A = {x, y}) → ran W = ℘B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860   ⊆ wss 3258   ifcif 3663  ℘cpw 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