New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  brsnsi2 GIF version

Theorem brsnsi2 5776
 Description: Binary relationship of an arbitrary set to a singleton in a singleton image. (Contributed by SF, 9-Mar-2015.)
Hypothesis
Ref Expression
brsnsi1.1 A V
Assertion
Ref Expression
brsnsi2 (B SI R{A} ↔ x(B = {x} xRA))
Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem brsnsi2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 brsi 4761 . 2 (B SI R{A} ↔ xy(B = {x} {A} = {y} xRy))
2 3anass 938 . . . . 5 ((B = {x} {A} = {y} xRy) ↔ (B = {x} ({A} = {y} xRy)))
32exbii 1582 . . . 4 (y(B = {x} {A} = {y} xRy) ↔ y(B = {x} ({A} = {y} xRy)))
4 19.42v 1905 . . . . 5 (y(B = {x} ({A} = {y} xRy)) ↔ (B = {x} y({A} = {y} xRy)))
5 brsnsi1.1 . . . . . . . . . . 11 A V
65sneqb 3876 . . . . . . . . . 10 ({A} = {y} ↔ A = y)
7 eqcom 2355 . . . . . . . . . 10 (A = yy = A)
86, 7bitri 240 . . . . . . . . 9 ({A} = {y} ↔ y = A)
98anbi1i 676 . . . . . . . 8 (({A} = {y} xRy) ↔ (y = A xRy))
109exbii 1582 . . . . . . 7 (y({A} = {y} xRy) ↔ y(y = A xRy))
11 breq2 4643 . . . . . . . 8 (y = A → (xRyxRA))
125, 11ceqsexv 2894 . . . . . . 7 (y(y = A xRy) ↔ xRA)
1310, 12bitri 240 . . . . . 6 (y({A} = {y} xRy) ↔ xRA)
1413anbi2i 675 . . . . 5 ((B = {x} y({A} = {y} xRy)) ↔ (B = {x} xRA))
154, 14bitri 240 . . . 4 (y(B = {x} ({A} = {y} xRy)) ↔ (B = {x} xRA))
163, 15bitri 240 . . 3 (y(B = {x} {A} = {y} xRy) ↔ (B = {x} xRA))
1716exbii 1582 . 2 (xy(B = {x} {A} = {y} xRy) ↔ x(B = {x} xRA))
181, 17bitri 240 1 (B SI R{A} ↔ x(B = {x} xRA))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737   class class class wbr 4639   SI csi 4720 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-si 4728 This theorem is referenced by:  brimage  5793  enpw1lem1  6061  nmembers1lem1  6268  nchoicelem11  6299
 Copyright terms: Public domain W3C validator