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

Theorem rnsnop 5075
 Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by set.mm contributors, 24-Jul-2004.)
Hypothesis
Ref Expression
rnsnop.1 A V
Assertion
Ref Expression
rnsnop ran {A, B} = {B}

Proof of Theorem rnsnop
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4640 . . . . . 6 (x{A, B}yx, y {A, B})
2 vex 2862 . . . . . . . . 9 x V
3 vex 2862 . . . . . . . . 9 y V
42, 3opex 4588 . . . . . . . 8 x, y V
54elsnc 3756 . . . . . . 7 (x, y {A, B} ↔ x, y = A, B)
6 opth 4602 . . . . . . 7 (x, y = A, B ↔ (x = A y = B))
75, 6bitri 240 . . . . . 6 (x, y {A, B} ↔ (x = A y = B))
81, 7bitri 240 . . . . 5 (x{A, B}y ↔ (x = A y = B))
98exbii 1582 . . . 4 (x x{A, B}yx(x = A y = B))
10 rnsnop.1 . . . . 5 A V
11 biidd 228 . . . . 5 (x = A → (y = By = B))
1210, 11ceqsexv 2894 . . . 4 (x(x = A y = B) ↔ y = B)
139, 12bitri 240 . . 3 (x x{A, B}yy = B)
14 elrn 4896 . . 3 (y ran {A, B} ↔ x x{A, B}y)
153elsnc 3756 . . 3 (y {B} ↔ y = B)
1613, 14, 153bitr4i 268 . 2 (y ran {A, B} ↔ y {B})
1716eqriv 2350 1 ran {A, B} = {B}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⟨cop 4561   class class class wbr 4639  ran crn 4773 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-br 4640  df-ima 4727  df-rn 4786 This theorem is referenced by:  op2nda  5076  fpr  5437  1cnc  6139
 Copyright terms: Public domain W3C validator