Theorem nenpw1pwlem1 6085

Description: Lemma for nenpw1pw 6087. Set up stratification. (Contributed by SF, 10-Mar-2015.)

Hypothesis

Ref	Expression
nenpw1pwlem1.1	⊢ S = {x ∈ A ∣ ¬ x ∈ (r ‘{x})}

Assertion

Ref	Expression
nenpw1pwlem1	⊢ (A ∈ V → S ∈ V)

Distinct variable groups:   x,A   x,r

Allowed substitution hints:   A(r)   S(x, r)   V(x, r)

Proof of Theorem nenpw1pwlem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nenpw1pwlem1.1	. . 3 ⊢ S = {x ∈ A ∣ ¬ x ∈ (r ‘{x})}
2		dfrab2 3531	. . 3 ⊢ {x ∈ A ∣ ¬ x ∈ (r ‘{x})} = ({x ∣ ¬ x ∈ (r ‘{x})} ∩ A)
3	1, 2	eqtri 2373	. 2 ⊢ S = ({x ∣ ¬ x ∈ (r ‘{x})} ∩ A)
4		vex 2863	. . . . . . 7 ⊢ x ∈ V
5	4	elcompl 3226	. . . . . 6 ⊢ (x ∈ ∼ ⋃1dom ( FullFun r ∩ S ) ↔ ¬ x ∈ ⋃1dom ( FullFun r ∩ S ))
6		eldm2 4900	. . . . . . . 8 ⊢ ({x} ∈ dom ( FullFun r ∩ S ) ↔ ∃y⟨{x}, y⟩ ∈ ( FullFun r ∩ S ))
7		elin 3220	. . . . . . . . . 10 ⊢ (⟨{x}, y⟩ ∈ ( FullFun r ∩ S ) ↔ (⟨{x}, y⟩ ∈ FullFun r ∧ ⟨{x}, y⟩ ∈ S ))
8		snex 4112	. . . . . . . . . . . . 13 ⊢ {x} ∈ V
9	8	brfullfun 5867	. . . . . . . . . . . 12 ⊢ ({x} FullFun ry ↔ (r ‘{x}) = y)
10		df-br 4641	. . . . . . . . . . . 12 ⊢ ({x} FullFun ry ↔ ⟨{x}, y⟩ ∈ FullFun r)
11		eqcom 2355	. . . . . . . . . . . 12 ⊢ ((r ‘{x}) = y ↔ y = (r ‘{x}))
12	9, 10, 11	3bitr3i 266	. . . . . . . . . . 11 ⊢ (⟨{x}, y⟩ ∈ FullFun r ↔ y = (r ‘{x}))
13		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
14	4, 13	opelssetsn 4761	. . . . . . . . . . 11 ⊢ (⟨{x}, y⟩ ∈ S ↔ x ∈ y)
15	12, 14	anbi12i 678	. . . . . . . . . 10 ⊢ ((⟨{x}, y⟩ ∈ FullFun r ∧ ⟨{x}, y⟩ ∈ S ) ↔ (y = (r ‘{x}) ∧ x ∈ y))
16	7, 15	bitri 240	. . . . . . . . 9 ⊢ (⟨{x}, y⟩ ∈ ( FullFun r ∩ S ) ↔ (y = (r ‘{x}) ∧ x ∈ y))
17	16	exbii 1582	. . . . . . . 8 ⊢ (∃y⟨{x}, y⟩ ∈ ( FullFun r ∩ S ) ↔ ∃y(y = (r ‘{x}) ∧ x ∈ y))
18	6, 17	bitri 240	. . . . . . 7 ⊢ ({x} ∈ dom ( FullFun r ∩ S ) ↔ ∃y(y = (r ‘{x}) ∧ x ∈ y))
19	4	eluni1 4174	. . . . . . 7 ⊢ (x ∈ ⋃1dom ( FullFun r ∩ S ) ↔ {x} ∈ dom ( FullFun r ∩ S ))
20		fvex 5340	. . . . . . . 8 ⊢ (r ‘{x}) ∈ V
21	20	clel3 2978	. . . . . . 7 ⊢ (x ∈ (r ‘{x}) ↔ ∃y(y = (r ‘{x}) ∧ x ∈ y))
22	18, 19, 21	3bitr4i 268	. . . . . 6 ⊢ (x ∈ ⋃1dom ( FullFun r ∩ S ) ↔ x ∈ (r ‘{x}))
23	5, 22	xchbinx 301	. . . . 5 ⊢ (x ∈ ∼ ⋃1dom ( FullFun r ∩ S ) ↔ ¬ x ∈ (r ‘{x}))
24	23	eqabi 2465	. . . 4 ⊢ ∼ ⋃1dom ( FullFun r ∩ S ) = {x ∣ ¬ x ∈ (r ‘{x})}
25		vex 2863	. . . . . . . . 9 ⊢ r ∈ V
26	25	fullfunex 5861	. . . . . . . 8 ⊢ FullFun r ∈ V
27		ssetex 4745	. . . . . . . 8 ⊢ S ∈ V
28	26, 27	inex 4106	. . . . . . 7 ⊢ ( FullFun r ∩ S ) ∈ V
29	28	dmex 5107	. . . . . 6 ⊢ dom ( FullFun r ∩ S ) ∈ V
30	29	uni1ex 4294	. . . . 5 ⊢ ⋃1dom ( FullFun r ∩ S ) ∈ V
31	30	complex 4105	. . . 4 ⊢ ∼ ⋃1dom ( FullFun r ∩ S ) ∈ V
32	24, 31	eqeltrri 2424	. . 3 ⊢ {x ∣ ¬ x ∈ (r ‘{x})} ∈ V
33		inexg 4101	. . 3 ⊢ (({x ∣ ¬ x ∈ (r ‘{x})} ∈ V ∧ A ∈ V) → ({x ∣ ¬ x ∈ (r ‘{x})} ∩ A) ∈ V)
34	32, 33	mpan 651	. 2 ⊢ (A ∈ V → ({x ∣ ¬ x ∈ (r ‘{x})} ∩ A) ∈ V)
35	3, 34	syl5eqel 2437	1 ⊢ (A ∈ V → S ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209  {csn 3738  ⋃₁cuni1 4134  ⟨cop 4562   class class class wbr 4640   S csset 4720  dom cdm 4773   ‘cfv 4782   FullFun cfullfun 5768

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-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-2nd 4798  df-fullfun 5769

This theorem is used by:  nenpw1pwlem2  6086