Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brsingle Structured version   Visualization version   GIF version

Theorem brsingle 32345
Description: The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brsingle.1 𝐴 ∈ V
brsingle.2 𝐵 ∈ V
Assertion
Ref Expression
brsingle (𝐴Singleton𝐵𝐵 = {𝐴})

Proof of Theorem brsingle
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 brsingle.1 . 2 𝐴 ∈ V
2 brsingle.2 . 2 𝐵 ∈ V
3 df-singleton 32290 . 2 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 brxp 5347 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 693 . 2 𝐴(V × V)𝐵
6 velsn 4386 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
71ideq 5476 . . 3 (𝑥 I 𝐴𝑥 = 𝐴)
86, 7bitr4i 269 . 2 (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
91, 2, 3, 5, 8brtxpsd3 32324 1 (𝐴Singleton𝐵𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2156  Vcvv 3391  {csn 4370   class class class wbr 4844   I cid 5218   × cxp 5309  Singletoncsingle 32266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-symdif 4042  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-eprel 5224  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7398  df-2nd 7399  df-txp 32282  df-singleton 32290
This theorem is referenced by:  elsingles  32346  fnsingle  32347  fvsingle  32348  brapply  32366  brsuccf  32369  funpartlem  32370
  Copyright terms: Public domain W3C validator