New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  enprmaplem5 GIF version

Theorem enprmaplem5 6080
 Description: Lemma for enprmap 6082. Establish that ℘B is a subset of the range of W. (Contributed by SF, 3-Mar-2015.)
Hypotheses
Ref Expression
enprmaplem5.1 W = (r (Am B) (r “ {x}))
enprmaplem5.2 R = (u B if(u p, x, y))
enprmaplem5.3 B V
Assertion
Ref Expression
enprmaplem5 ((xy A = {x, y}) → B 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.
StepHypRef Expression
1 vex 2862 . . . 4 p V
21elpw 3728 . . 3 (p Bp B)
3 ifeqor 3699 . . . . . . . . . . . . . 14 ( if(u p, x, y) = x if(u p, x, y) = y)
4 vex 2862 . . . . . . . . . . . . . . . 16 x V
5 vex 2862 . . . . . . . . . . . . . . . 16 y V
64, 5ifex 3720 . . . . . . . . . . . . . . 15 if(u p, x, y) V
76elpr 3751 . . . . . . . . . . . . . 14 ( if(u p, x, y) {x, y} ↔ ( if(u p, x, y) = x if(u p, x, y) = y))
83, 7mpbir 200 . . . . . . . . . . . . 13 if(u p, x, y) {x, y}
9 id 19 . . . . . . . . . . . . 13 (A = {x, y} → A = {x, y})
108, 9syl5eleqr 2440 . . . . . . . . . . . 12 (A = {x, y} → if(u p, x, y) A)
1110ralrimivw 2698 . . . . . . . . . . 11 (A = {x, y} → u B if(u p, x, y) A)
12 enprmaplem5.2 . . . . . . . . . . . 12 R = (u B if(u p, x, y))
1312fmpt 5692 . . . . . . . . . . 11 (u B if(u p, x, y) AR:B–→A)
1411, 13sylib 188 . . . . . . . . . 10 (A = {x, y} → R:B–→A)
15 prex 4112 . . . . . . . . . . . 12 {x, y} V
16 eleq1 2413 . . . . . . . . . . . 12 (A = {x, y} → (A V ↔ {x, y} V))
1715, 16mpbiri 224 . . . . . . . . . . 11 (A = {x, y} → A V)
18 enprmaplem5.3 . . . . . . . . . . . 12 B V
1912, 18enprmaplem4 6079 . . . . . . . . . . . 12 R V
20 elmapg 6012 . . . . . . . . . . . 12 ((A V B V R V) → (R (Am B) ↔ R:B–→A))
2118, 19, 20mp3an23 1269 . . . . . . . . . . 11 (A V → (R (Am B) ↔ R:B–→A))
2217, 21syl 15 . . . . . . . . . 10 (A = {x, y} → (R (Am B) ↔ R:B–→A))
2314, 22mpbird 223 . . . . . . . . 9 (A = {x, y} → R (Am B))
24233ad2ant2 977 . . . . . . . 8 ((xy A = {x, y} p B) → R (Am B))
25 cnveq 4886 . . . . . . . . . 10 (r = Rr = R)
2625imaeq1d 4941 . . . . . . . . 9 (r = R → (r “ {x}) = (R “ {x}))
27 enprmaplem5.1 . . . . . . . . 9 W = (r (Am B) (r “ {x}))
2819cnvex 5102 . . . . . . . . . 10 R V
29 snex 4111 . . . . . . . . . 10 {x} V
3028, 29imaex 4747 . . . . . . . . 9 (R “ {x}) V
3126, 27, 30fvmpt 5700 . . . . . . . 8 (R (Am B) → (WR) = (R “ {x}))
3224, 31syl 15 . . . . . . 7 ((xy A = {x, y} p B) → (WR) = (R “ {x}))
33 eliniseg 5020 . . . . . . . . 9 (z (R “ {x}) ↔ zRx)
34 breldm 4911 . . . . . . . . . . . . 13 (zRxz dom R)
3512fnmpt 5689 . . . . . . . . . . . . . . 15 (u B if(u p, x, y) V → R Fn B)
366a1i 10 . . . . . . . . . . . . . . 15 (u B → if(u p, x, y) V)
3735, 36mprg 2683 . . . . . . . . . . . . . 14 R Fn B
38 fndm 5182 . . . . . . . . . . . . . 14 (R Fn B → dom R = B)
3937, 38ax-mp 8 . . . . . . . . . . . . 13 dom R = B
4034, 39syl6eleq 2443 . . . . . . . . . . . 12 (zRxz B)
41 fnbrfvb 5358 . . . . . . . . . . . . . . 15 ((R Fn B z B) → ((Rz) = xzRx))
4237, 41mpan 651 . . . . . . . . . . . . . 14 (z B → ((Rz) = xzRx))
4342biimprd 214 . . . . . . . . . . . . 13 (z B → (zRx → (Rz) = x))
4443com12 27 . . . . . . . . . . . 12 (zRx → (z B → (Rz) = x))
4540, 44jcai 522 . . . . . . . . . . 11 (zRx → (z B (Rz) = x))
46 eleq1 2413 . . . . . . . . . . . . . . . . 17 (u = z → (u pz p))
4746ifbid 3680 . . . . . . . . . . . . . . . 16 (u = z → if(u p, x, y) = if(z p, x, y))
484, 5ifex 3720 . . . . . . . . . . . . . . . 16 if(z p, x, y) V
4947, 12, 48fvmpt 5700 . . . . . . . . . . . . . . 15 (z B → (Rz) = if(z p, x, y))
5049eqeq1d 2361 . . . . . . . . . . . . . 14 (z B → ((Rz) = x ↔ if(z p, x, y) = x))
5150biimpd 198 . . . . . . . . . . . . 13 (z B → ((Rz) = x → if(z p, x, y) = x))
5251imp 418 . . . . . . . . . . . 12 ((z B (Rz) = x) → if(z p, x, y) = x)
53 simpl1 958 . . . . . . . . . . . . . . 15 (((xy A = {x, y} p B) if(z p, x, y) = x) → xy)
54 df-ne 2518 . . . . . . . . . . . . . . 15 (xy ↔ ¬ x = y)
5553, 54sylib 188 . . . . . . . . . . . . . 14 (((xy A = {x, y} p B) if(z p, x, y) = x) → ¬ x = y)
56 iffalse 3669 . . . . . . . . . . . . . . . . . . 19 z p → if(z p, x, y) = y)
5756eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 z p → (x = if(z p, x, y) ↔ x = y))
5857biimpd 198 . . . . . . . . . . . . . . . . 17 z p → (x = if(z p, x, y) → x = y))
5958com12 27 . . . . . . . . . . . . . . . 16 (x = if(z p, x, y) → (¬ z px = y))
6059eqcoms 2356 . . . . . . . . . . . . . . 15 ( if(z p, x, y) = x → (¬ z px = y))
6160adantl 452 . . . . . . . . . . . . . 14 (((xy A = {x, y} p B) if(z p, x, y) = x) → (¬ z px = y))
6255, 61mt3d 117 . . . . . . . . . . . . 13 (((xy A = {x, y} p B) if(z p, x, y) = x) → z p)
6362ex 423 . . . . . . . . . . . 12 ((xy A = {x, y} p B) → ( if(z p, x, y) = xz p))
6452, 63syl5 28 . . . . . . . . . . 11 ((xy A = {x, y} p B) → ((z B (Rz) = x) → z p))
6545, 64syl5 28 . . . . . . . . . 10 ((xy A = {x, y} p B) → (zRxz p))
66 ssel2 3268 . . . . . . . . . . . . . . 15 ((p B z p) → z B)
67663ad2antl3 1119 . . . . . . . . . . . . . 14 (((xy A = {x, y} p B) z p) → z B)
6867, 49syl 15 . . . . . . . . . . . . 13 (((xy A = {x, y} p B) z p) → (Rz) = if(z p, x, y))
69 iftrue 3668 . . . . . . . . . . . . . 14 (z p → if(z p, x, y) = x)
7069adantl 452 . . . . . . . . . . . . 13 (((xy A = {x, y} p B) z p) → if(z p, x, y) = x)
7168, 70eqtrd 2385 . . . . . . . . . . . 12 (((xy A = {x, y} p B) z p) → (Rz) = x)
7267, 42syl 15 . . . . . . . . . . . 12 (((xy A = {x, y} p B) z p) → ((Rz) = xzRx))
7371, 72mpbid 201 . . . . . . . . . . 11 (((xy A = {x, y} p B) z p) → zRx)
7473ex 423 . . . . . . . . . 10 ((xy A = {x, y} p B) → (z pzRx))
7565, 74impbid 183 . . . . . . . . 9 ((xy A = {x, y} p B) → (zRxz p))
7633, 75syl5bb 248 . . . . . . . 8 ((xy A = {x, y} p B) → (z (R “ {x}) ↔ z p))
7776eqrdv 2351 . . . . . . 7 ((xy A = {x, y} p B) → (R “ {x}) = p)
7832, 77eqtrd 2385 . . . . . 6 ((xy A = {x, y} p B) → (WR) = p)
7927enprmaplem2 6077 . . . . . . . 8 W Fn (Am B)
80 fnbrfvb 5358 . . . . . . . 8 ((W Fn (Am B) R (Am B)) → ((WR) = pRWp))
8179, 80mpan 651 . . . . . . 7 (R (Am B) → ((WR) = pRWp))
8224, 81syl 15 . . . . . 6 ((xy A = {x, y} p B) → ((WR) = pRWp))
8378, 82mpbid 201 . . . . 5 ((xy A = {x, y} p B) → RWp)
84833expia 1153 . . . 4 ((xy A = {x, y}) → (p BRWp))
85 brelrn 4960 . . . 4 (RWpp ran W)
8684, 85syl6 29 . . 3 ((xy A = {x, y}) → (p Bp ran W))
872, 86syl5bi 208 . 2 ((xy A = {x, y}) → (p Bp ran W))
8887ssrdv 3278 1 ((xy A = {x, y}) → B ran W)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  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:  enprmaplem6  6081
 Copyright terms: Public domain W3C validator