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Theorem enprmaplem6 6081
 Description: Lemma for enprmap 6082. The range of W is ℘B. (Contributed by SF, 3-Mar-2015.)
Hypotheses
Ref Expression
enprmaplem6.1 W = (r (Am B) (r “ {x}))
enprmaplem6.2 B V
Assertion
Ref Expression
enprmaplem6 ((xy 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.
StepHypRef Expression
1 breldm 4911 . . . . . . . 8 (sWps dom W)
2 enprmaplem6.1 . . . . . . . . . 10 W = (r (Am B) (r “ {x}))
32enprmaplem2 6077 . . . . . . . . 9 W Fn (Am B)
4 fndm 5182 . . . . . . . . 9 (W Fn (Am B) → dom W = (Am B))
53, 4ax-mp 8 . . . . . . . 8 dom W = (Am B)
61, 5syl6eleq 2443 . . . . . . 7 (sWps (Am B))
7 fnbrfvb 5358 . . . . . . . . 9 ((W Fn (Am B) s (Am B)) → ((Ws) = psWp))
83, 6, 7sylancr 644 . . . . . . . 8 (sWp → ((Ws) = psWp))
98ibir 233 . . . . . . 7 (sWp → (Ws) = p)
106, 9jca 518 . . . . . 6 (sWp → (s (Am B) (Ws) = p))
11 cnveq 4886 . . . . . . . . . . . . 13 (r = sr = s)
1211imaeq1d 4941 . . . . . . . . . . . 12 (r = s → (r “ {x}) = (s “ {x}))
13 vex 2862 . . . . . . . . . . . . . 14 s V
1413cnvex 5102 . . . . . . . . . . . . 13 s V
15 snex 4111 . . . . . . . . . . . . 13 {x} V
1614, 15imaex 4747 . . . . . . . . . . . 12 (s “ {x}) V
1712, 2, 16fvmpt 5700 . . . . . . . . . . 11 (s (Am B) → (Ws) = (s “ {x}))
1817eqeq1d 2361 . . . . . . . . . 10 (s (Am B) → ((Ws) = p ↔ (s “ {x}) = p))
19183ad2ant3 978 . . . . . . . . 9 ((xy A = {x, y} s (Am B)) → ((Ws) = p ↔ (s “ {x}) = p))
20 imassrn 5009 . . . . . . . . . . . 12 (s “ {x}) ran s
21 df-dm 4787 . . . . . . . . . . . . 13 dom s = ran s
22 elmapi 6016 . . . . . . . . . . . . . 14 (s (Am B) → s:B–→A)
23 fdm 5226 . . . . . . . . . . . . . 14 (s:B–→A → dom s = B)
24 eqimss 3323 . . . . . . . . . . . . . 14 (dom s = B → dom s B)
2522, 23, 243syl 18 . . . . . . . . . . . . 13 (s (Am B) → dom s B)
2621, 25syl5eqssr 3316 . . . . . . . . . . . 12 (s (Am B) → ran s B)
2720, 26syl5ss 3283 . . . . . . . . . . 11 (s (Am B) → (s “ {x}) B)
28273ad2ant3 978 . . . . . . . . . 10 ((xy A = {x, y} s (Am B)) → (s “ {x}) B)
29 sseq1 3292 . . . . . . . . . 10 ((s “ {x}) = p → ((s “ {x}) Bp B))
3028, 29syl5ibcom 211 . . . . . . . . 9 ((xy A = {x, y} s (Am B)) → ((s “ {x}) = pp B))
3119, 30sylbid 206 . . . . . . . 8 ((xy A = {x, y} s (Am B)) → ((Ws) = pp B))
32313expia 1153 . . . . . . 7 ((xy A = {x, y}) → (s (Am B) → ((Ws) = pp B)))
3332imp3a 420 . . . . . 6 ((xy A = {x, y}) → ((s (Am B) (Ws) = p) → p B))
3410, 33syl5 28 . . . . 5 ((xy A = {x, y}) → (sWpp B))
3534exlimdv 1636 . . . 4 ((xy A = {x, y}) → (s sWpp B))
36 elrn 4896 . . . 4 (p ran Ws sWp)
37 vex 2862 . . . . 5 p V
3837elpw 3728 . . . 4 (p Bp B)
3935, 36, 383imtr4g 261 . . 3 ((xy A = {x, y}) → (p ran Wp B))
4039ssrdv 3278 . 2 ((xy 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
432, 41, 42enprmaplem5 6080 . 2 ((xy A = {x, y}) → B ran W)
4440, 43eqssd 3289 1 ((xy A = {x, y}) → ran W = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ⊆ wss 3257   ifcif 3662  ℘cpw 3722  {csn 3737  {cpr 3738   class class class wbr 4639   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773   Fn wfn 4776  –→wf 4777   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑m cmap 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001 This theorem is referenced by:  enprmap  6082
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