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.

Step	Hyp	Ref	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} ∈ ℘1x ↔ z ∈ x)
5	4	anbi2i 675	. . . . . . . 8 ⊢ ((u ∈ p ∧ {z} ∈ ℘1x) ↔ (u ∈ p ∧ z ∈ x))
6	3, 5	bitri 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
9	8	elcompl 3225	. . . . . . . . 9 ⊢ (u ∈ ∼ p ↔ ¬ u ∈ p)
10		snelpw1 4146	. . . . . . . . 9 ⊢ ({z} ∈ ℘1y ↔ z ∈ y)
11	9, 10	anbi12i 678	. . . . . . . 8 ⊢ ((u ∈ ∼ p ∧ {z} ∈ ℘1y) ↔ (¬ u ∈ p ∧ z ∈ y))
12	7, 11	bitri 240	. . . . . . 7 ⊢ (⟨u, {z}⟩ ∈ ( ∼ p × ℘1y) ↔ (¬ u ∈ p ∧ z ∈ y))
13	6, 12	orbi12i 507	. . . . . 6 ⊢ ((⟨u, {z}⟩ ∈ (p × ℘1x) ∨ ⟨u, {z}⟩ ∈ ( ∼ p × ℘1y)) ↔ ((u ∈ p ∧ z ∈ x) ∨ (¬ u ∈ p ∧ z ∈ y)))
14	2, 13	bitri 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)))
17	14, 15, 16	3bitr4i 268	. . . 4 ⊢ (⟨{z}, u⟩ ∈ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ z ∈ if(u ∈ p, x, y))
18	17	releqmpt 5808	. . 3 ⊢ ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) = (u ∈ B ↦ if(u ∈ p, x, y))
19	1, 18	eqtr4i 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
23	22	pw1ex 4303	. . . . . 6 ⊢ ℘1x ∈ V
24	21, 23	xpex 5115	. . . . 5 ⊢ (p × ℘1x) ∈ V
25	21	complex 4104	. . . . . 6 ⊢ ∼ p ∈ V
26		vex 2862	. . . . . . 7 ⊢ y ∈ V
27	26	pw1ex 4303	. . . . . 6 ⊢ ℘1y ∈ V
28	25, 27	xpex 5115	. . . . 5 ⊢ ( ∼ p × ℘1y) ∈ V
29	24, 28	unex 4106	. . . 4 ⊢ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V
30	29	cnvex 5102	. . 3 ⊢ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V
31	20, 30	mptexlem 5810	. 2 ⊢ ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) ∈ V
32	19, 31	eqeltri 2423	1 ⊢ R ∈ V

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  1_cc1c 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-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 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