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Theorem enprmaplem4 6079
 Description: Lemma for enprmap 6082. More stratification condition setup. (Contributed by SF, 3-Mar-2015.)
Hypotheses
Ref Expression
enprmaplem4.1 R = (u B if(u p, x, y))
enprmaplem4.2 B V
Assertion
Ref Expression
enprmaplem4 R V
Distinct variable groups:   u,B   u,p   x,u   y,u
Allowed substitution hints:   B(x,y,p)   R(x,y,u,p)

Proof of Theorem enprmaplem4
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 enprmaplem4.1 . . 3 R = (u B if(u p, x, y))
2 elun 3220 . . . . . 6 (u, {z} ((p × 1x) ∪ ( ∼ p × 1y)) ↔ (u, {z} (p × 1x) u, {z} ( ∼ p × 1y)))
3 opelxp 4811 . . . . . . . 8 (u, {z} (p × 1x) ↔ (u p {z} 1x))
4 snelpw1 4146 . . . . . . . . 9 ({z} 1xz x)
54anbi2i 675 . . . . . . . 8 ((u p {z} 1x) ↔ (u p z x))
63, 5bitri 240 . . . . . . 7 (u, {z} (p × 1x) ↔ (u p z x))
7 opelxp 4811 . . . . . . . 8 (u, {z} ( ∼ p × 1y) ↔ (u p {z} 1y))
8 vex 2862 . . . . . . . . . 10 u V
98elcompl 3225 . . . . . . . . 9 (u p ↔ ¬ u p)
10 snelpw1 4146 . . . . . . . . 9 ({z} 1yz y)
119, 10anbi12i 678 . . . . . . . 8 ((u p {z} 1y) ↔ (¬ u p z y))
127, 11bitri 240 . . . . . . 7 (u, {z} ( ∼ p × 1y) ↔ (¬ u p z y))
136, 12orbi12i 507 . . . . . 6 ((u, {z} (p × 1x) u, {z} ( ∼ p × 1y)) ↔ ((u p z x) u p z y)))
142, 13bitri 240 . . . . 5 (u, {z} ((p × 1x) ∪ ( ∼ p × 1y)) ↔ ((u p z x) u p z y)))
15 opelcnv 4893 . . . . 5 ({z}, u ((p × 1x) ∪ ( ∼ p × 1y)) ↔ u, {z} ((p × 1x) ∪ ( ∼ p × 1y)))
16 elif 3696 . . . . 5 (z if(u p, x, y) ↔ ((u p z x) u p z y)))
1714, 15, 163bitr4i 268 . . . 4 ({z}, u ((p × 1x) ∪ ( ∼ p × 1y)) ↔ z if(u p, x, y))
1817releqmpt 5808 . . 3 ((B × V) ∩ ∼ (( Ins3 S Ins2 ((p × 1x) ∪ ( ∼ p × 1y))) “ 1c)) = (u B if(u p, x, y))
191, 18eqtr4i 2376 . 2 R = ((B × V) ∩ ∼ (( Ins3 S Ins2 ((p × 1x) ∪ ( ∼ p × 1y))) “ 1c))
20 enprmaplem4.2 . . 3 B V
21 vex 2862 . . . . . 6 p V
22 vex 2862 . . . . . . 7 x V
2322pw1ex 4303 . . . . . 6 1x V
2421, 23xpex 5115 . . . . 5 (p × 1x) V
2521complex 4104 . . . . . 6 p V
26 vex 2862 . . . . . . 7 y V
2726pw1ex 4303 . . . . . 6 1y V
2825, 27xpex 5115 . . . . 5 ( ∼ p × 1y) V
2924, 28unex 4106 . . . 4 ((p × 1x) ∪ ( ∼ p × 1y)) V
3029cnvex 5102 . . 3 ((p × 1x) ∪ ( ∼ p × 1y)) V
3120, 30mptexlem 5810 . 2 ((B × V) ∩ ∼ (( Ins3 S Ins2 ((p × 1x) ∪ ( ∼ p × 1y))) “ 1c)) V
3219, 31eqeltri 2423 1 R V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208   ⊕ csymdif 3209   ifcif 3662  {csn 3737  1cc1c 4134  ℘1cpw1 4135  ⟨cop 4561   S csset 4719   “ cima 4722   × cxp 4770  ◡ccnv 4771   ↦ cmpt 5651   Ins2 cins2 5749   Ins3 cins3 5751 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-xp 4784  df-cnv 4785  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752 This theorem is referenced by:  enprmaplem5  6080
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