Theorem dfsset2 4743

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 2862	. . . 4 |- x e. _V
2		vex 2862	. . . 4 |- y e. _V
3		opkelssetkg 4268	. . . 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 4626	. 2 |- {<.x, y>. | <<x, y>> e. Sk } = {<.x, y>. | x C_ y}
6		setconslem4 4734	. 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 4725	. 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 2859   ∼ ccompl 3205   \ cdif 3206   u. cun 3207   i^i cin 3208   (+) csymdif 3209   C_ wss 3257  {csn 3737  <<copk 4057  ⋃₁cuni1 4133  1_cc1c 4134  ~P₁ cpw1 4135   X._k cxpk 4174  `'_kccnvk 4175   Ins2_k cins2k 4176   Ins3_k cins3k 4177  "_kcimak 4179   o._k ccomk 4180   SI_k csik 4181  Image_kcimagek 4182   S_k cssetk 4183   _I_k_k cidk 4184   Nn cnnc 4373  0_cc0c 4374  {copab 4622   S csset 4719

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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-sset 4725

This theorem is referenced by:  ssetex  4744