Theorem enprmaplem4 6080

Description: Lemma for enprmap 6083. 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 3221	. . . . . 6 ⊢ (⟨u, {z}⟩ ∈ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ (⟨u, {z}⟩ ∈ (p × ℘1x) ∨ ⟨u, {z}⟩ ∈ ( ∼ p × ℘1y)))
3		opelxp 4812	. . . . . . . 8 ⊢ (⟨u, {z}⟩ ∈ (p × ℘1x) ↔ (u ∈ p ∧ {z} ∈ ℘1x))
4		snelpw1 4147	. . . . . . . . 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 4812	. . . . . . . 8 ⊢ (⟨u, {z}⟩ ∈ ( ∼ p × ℘1y) ↔ (u ∈ ∼ p ∧ {z} ∈ ℘1y))
8		vex 2863	. . . . . . . . . 10 ⊢ u ∈ V
9	8	elcompl 3226	. . . . . . . . 9 ⊢ (u ∈ ∼ p ↔ ¬ u ∈ p)
10		snelpw1 4147	. . . . . . . . 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 4894	. . . . 5 ⊢ (⟨{z}, u⟩ ∈ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ↔ ⟨u, {z}⟩ ∈ ((p × ℘1x) ∪ ( ∼ p × ℘1y)))
16		elif 3697	. . . . 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 5809	. . 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 2863	. . . . . 6 ⊢ p ∈ V
22		vex 2863	. . . . . . 7 ⊢ x ∈ V
23	22	pw1ex 4304	. . . . . 6 ⊢ ℘1x ∈ V
24	21, 23	xpex 5116	. . . . 5 ⊢ (p × ℘1x) ∈ V
25	21	complex 4105	. . . . . 6 ⊢ ∼ p ∈ V
26		vex 2863	. . . . . . 7 ⊢ y ∈ V
27	26	pw1ex 4304	. . . . . 6 ⊢ ℘1y ∈ V
28	25, 27	xpex 5116	. . . . 5 ⊢ ( ∼ p × ℘1y) ∈ V
29	24, 28	unex 4107	. . . 4 ⊢ ((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V
30	29	cnvex 5103	. . 3 ⊢ ◡((p × ℘1x) ∪ ( ∼ p × ℘1y)) ∈ V
31	20, 30	mptexlem 5811	. 2 ⊢ ((B × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ◡((p × ℘1x) ∪ ( ∼ p × ℘1y))) “ 1c)) ∈ V
32	19, 31	eqeltri 2423	1 ⊢ R ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209   ⊕ csymdif 3210   ifcif 3663  {csn 3738  1_cc1c 4135  ℘₁cpw1 4136  ⟨cop 4562   S csset 4720   “ cima 4723   × cxp 4771  ^◡ccnv 4772   ↦ cmpt 5652   Ins2 cins2 5750   Ins3 cins3 5752

This proof depends on 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 proof 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-xp 4785  df-cnv 4786  df-2nd 4798  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753

This theorem is used by:  enprmaplem5  6081