Theorem dfsset2 4744

Description: Express the S relationship via the set construction functors. (Contributed by SF, 7-Jan-2015.)

Assertion

Ref

Expression

dfsset2

|- S = ⋃1⋃1((((_V X.k _V) X.k _V) i^i `'k ∼ (( Ins3k SIk SIk Sk (+) Ins2k ( Ins3k ( Sk o.k SIk `'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))) u. Ins2k (( Ins2k Sk i^i Ins3k SIk ∼ (( Ins2k Sk (+) Ins3k ((`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V))) o.k Sk ) u. ({{0c}} X.k _V)))"k~P1 ~P1 1c))"k~P1 ~P1 1c)))"k~P1 ~P1 ~P1 ~P1 1c))"k Sk )

Proof of Theorem dfsset2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 |- x e. _V
2		vex 2863	. . . 4 |- y e. _V
3		opkelssetkg 4269	. . . 4 |- ((x e. _V /\ y e. _V) -> (<<x, y>> e. Sk <-> x C_ y))
4	1, 2, 3	mp2an 653	. . 3 |- (<<x, y>> e. Sk <-> x C_ y)
5	4	opabbii 4627	. 2 |- {<.x, y>. | <<x, y>> e. Sk } = {<.x, y>. | x C_ y}
6		setconslem4 4735	. 2 |- ⋃1⋃1((((_V X.k _V) X.k _V) i^i `'k ∼ (( Ins3k SIk SIk Sk (+) Ins2k ( Ins3k ( Sk o.k SIk `'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))) u. Ins2k (( Ins2k Sk i^i Ins3k SIk ∼ (( Ins2k Sk (+) Ins3k ((`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V))) o.k Sk ) u. ({{0c}} X.k _V)))"k~P1 ~P1 1c))"k~P1 ~P1 1c)))"k~P1 ~P1 ~P1 ~P1 1c))"k Sk ) = {<.x, y>. | <<x, y>> e. Sk }
7		df-sset 4726	. 2 |- S = {<.x, y>. | x C_ y}
8	5, 6, 7	3eqtr4ri 2384	1 |- S = ⋃1⋃1((((_V X.k _V) X.k _V) i^i `'k ∼ (( Ins3k SIk SIk Sk (+) Ins2k ( Ins3k ( Sk o.k SIk `'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V)))) u. Ins2k (( Ins2k Sk i^i Ins3k SIk ∼ (( Ins2k Sk (+) Ins3k ((`'kImagek((Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) i^i ( Nn X.k _V)) u. ( _I_kk i^i ( ∼ Nn X.k _V))) o.k Sk ) u. ({{0c}} X.k _V)))"k~P1 ~P1 1c))"k~P1 ~P1 1c)))"k~P1 ~P1 ~P1 ~P1 1c))"k Sk )

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   u. cun 3208   i^i cin 3209   (+) csymdif 3210   C_ wss 3258  {csn 3738  <<copk 4058  ⋃₁cuni1 4134  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175  `'_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   o._k ccomk 4181   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   _I_k_k cidk 4185   Nn cnnc 4374  0_cc0c 4375  {copab 4623   S csset 4720

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-sset 4726

This theorem is referenced by:  ssetex  4745