Theorem brsnsi 5774

Description: Binary relationship of singletons in a singleton image. (Contributed by SF, 9-Feb-2015.)

Hypotheses

Ref	Expression
brsnsi.1	|- A e. _V
brsnsi.2	|- B e. _V

Assertion

Ref	Expression
brsnsi	|- ({A} SI R{B} <-> ARB)

Proof of Theorem brsnsi

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

Step	Hyp	Ref	Expression
1		snex 4112	. . 3 |- {A} e. _V
2		snex 4112	. . 3 |- {B} e. _V
3		eqeq1 2359	. . . . . 6 |- (z = {A} -> (z = {x} <-> {A} = {x}))
4		eqcom 2355	. . . . . . 7 |- ({A} = {x} <-> {x} = {A})
5		vex 2863	. . . . . . . 8 |- x e. _V
6	5	sneqb 3877	. . . . . . 7 |- ({x} = {A} <-> x = A)
7	4, 6	bitri 240	. . . . . 6 |- ({A} = {x} <-> x = A)
8	3, 7	syl6bb 252	. . . . 5 |- (z = {A} -> (z = {x} <-> x = A))
9	8	3anbi1d 1256	. . . 4 |- (z = {A} -> ((z = {x} /\ w = {y} /\ xRy) <-> (x = A /\ w = {y} /\ xRy)))
10	9	2exbidv 1628	. . 3 |- (z = {A} -> (E.xE.y(z = {x} /\ w = {y} /\ xRy) <-> E.xE.y(x = A /\ w = {y} /\ xRy)))
11		eqeq1 2359	. . . . . 6 |- (w = {B} -> (w = {y} <-> {B} = {y}))
12		eqcom 2355	. . . . . . 7 |- ({B} = {y} <-> {y} = {B})
13		vex 2863	. . . . . . . 8 |- y e. _V
14	13	sneqb 3877	. . . . . . 7 |- ({y} = {B} <-> y = B)
15	12, 14	bitri 240	. . . . . 6 |- ({B} = {y} <-> y = B)
16	11, 15	syl6bb 252	. . . . 5 |- (w = {B} -> (w = {y} <-> y = B))
17	16	3anbi2d 1257	. . . 4 |- (w = {B} -> ((x = A /\ w = {y} /\ xRy) <-> (x = A /\ y = B /\ xRy)))
18	17	2exbidv 1628	. . 3 |- (w = {B} -> (E.xE.y(x = A /\ w = {y} /\ xRy) <-> E.xE.y(x = A /\ y = B /\ xRy)))
19		df-si 4729	. . 3 |- SI R = {<.z, w>. | E.xE.y(z = {x} /\ w = {y} /\ xRy)}
20	1, 2, 10, 18, 19	brab 4710	. 2 |- ({A} SI R{B} <-> E.xE.y(x = A /\ y = B /\ xRy))
21		brsnsi.1	. . 3 |- A e. _V
22		brsnsi.2	. . 3 |- B e. _V
23		breq1 4643	. . 3 |- (x = A -> (xRy <-> ARy))
24		breq2 4644	. . 3 |- (y = B -> (ARy <-> ARB))
25	21, 22, 23, 24	ceqsex2v 2897	. 2 |- (E.xE.y(x = A /\ y = B /\ xRy) <-> ARB)
26	20, 25	bitri 240	1 |- ({A} SI R{B} <-> ARB)

Syntax hints:   <-> wb 176   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  {csn 3738   class class class wbr 4640   SI csi 4721

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-si 4729

This theorem is referenced by:  opsnelsi  5775  pw1fnex  5853  tcfnex  6245