Theorem nenpw1pwlem1 6085

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

Hypothesis

Ref	Expression
nenpw1pwlem1.1	|- S = {x e. A | -. x e. (r` {x})}

Assertion

Ref	Expression
nenpw1pwlem1	|- (A e. V -> S e. _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 e. A | -. x e. (r` {x})}
2		dfrab2 3531	. . 3 |- {x e. A | -. x e. (r` {x})} = ({x | -. x e. (r` {x})} i^i A)
3	1, 2	eqtri 2373	. 2 |- S = ({x | -. x e. (r` {x})} i^i A)
4		vex 2863	. . . . . . 7 |- x e. _V
5	4	elcompl 3226	. . . . . 6 |- (x e. ∼ ⋃1dom ( FullFun r i^i S ) <-> -. x e. ⋃1dom ( FullFun r i^i S ))
6		eldm2 4900	. . . . . . . 8 |- ({x} e. dom ( FullFun r i^i S ) <-> E.y<.{x}, y>. e. ( FullFun r i^i S ))
7		elin 3220	. . . . . . . . . 10 |- (<.{x}, y>. e. ( FullFun r i^i S ) <-> (<.{x}, y>. e. FullFun r /\ <.{x}, y>. e. S ))
8		snex 4112	. . . . . . . . . . . . 13 |- {x} e. _V
9	8	brfullfun 5867	. . . . . . . . . . . 12 |- ({x} FullFun ry <-> (r` {x}) = y)
10		df-br 4641	. . . . . . . . . . . 12 |- ({x} FullFun ry <-> <.{x}, y>. e. FullFun r)
11		eqcom 2355	. . . . . . . . . . . 12 |- ((r` {x}) = y <-> y = (r` {x}))
12	9, 10, 11	3bitr3i 266	. . . . . . . . . . 11 |- (<.{x}, y>. e. FullFun r <-> y = (r` {x}))
13		vex 2863	. . . . . . . . . . . 12 |- y e. _V
14	4, 13	opelssetsn 4761	. . . . . . . . . . 11 |- (<.{x}, y>. e. S <-> x e. y)
15	12, 14	anbi12i 678	. . . . . . . . . 10 |- ((<.{x}, y>. e. FullFun r /\ <.{x}, y>. e. S ) <-> (y = (r` {x}) /\ x e. y))
16	7, 15	bitri 240	. . . . . . . . 9 |- (<.{x}, y>. e. ( FullFun r i^i S ) <-> (y = (r` {x}) /\ x e. y))
17	16	exbii 1582	. . . . . . . 8 |- (E.y<.{x}, y>. e. ( FullFun r i^i S ) <-> E.y(y = (r` {x}) /\ x e. y))
18	6, 17	bitri 240	. . . . . . 7 |- ({x} e. dom ( FullFun r i^i S ) <-> E.y(y = (r` {x}) /\ x e. y))
19	4	eluni1 4174	. . . . . . 7 |- (x e. ⋃1dom ( FullFun r i^i S ) <-> {x} e. dom ( FullFun r i^i S ))
20		fvex 5340	. . . . . . . 8 |- (r` {x}) e. _V
21	20	clel3 2978	. . . . . . 7 |- (x e. (r` {x}) <-> E.y(y = (r` {x}) /\ x e. y))
22	18, 19, 21	3bitr4i 268	. . . . . 6 |- (x e. ⋃1dom ( FullFun r i^i S ) <-> x e. (r` {x}))
23	5, 22	xchbinx 301	. . . . 5 |- (x e. ∼ ⋃1dom ( FullFun r i^i S ) <-> -. x e. (r` {x}))
24	23	abbi2i 2465	. . . 4 |- ∼ ⋃1dom ( FullFun r i^i S ) = {x | -. x e. (r` {x})}
25		vex 2863	. . . . . . . . 9 |- r e. _V
26	25	fullfunex 5861	. . . . . . . 8 |- FullFun r e. _V
27		ssetex 4745	. . . . . . . 8 |- S e. _V
28	26, 27	inex 4106	. . . . . . 7 |- ( FullFun r i^i S ) e. _V
29	28	dmex 5107	. . . . . 6 |- dom ( FullFun r i^i S ) e. _V
30	29	uni1ex 4294	. . . . 5 |- ⋃1dom ( FullFun r i^i S ) e. _V
31	30	complex 4105	. . . 4 |- ∼ ⋃1dom ( FullFun r i^i S ) e. _V
32	24, 31	eqeltrri 2424	. . 3 |- {x | -. x e. (r` {x})} e. _V
33		inexg 4101	. . 3 |- (({x | -. x e. (r` {x})} e. _V /\ A e. V) -> ({x | -. x e. (r` {x})} i^i A) e. _V)
34	32, 33	mpan 651	. 2 |- (A e. V -> ({x | -. x e. (r` {x})} i^i A) e. _V)
35	3, 34	syl5eqel 2437	1 |- (A e. V -> S e. _V)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2619  _Vcvv 2860   ∼ ccompl 3206   i^i 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 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 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 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 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 referenced by:  nenpw1pwlem2  6086