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Theorem brsingle 32351
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 32296 . 2 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 brxp 5304 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 993 . 2 𝐴(V × V)𝐵
6 velsn 4337 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
71ideq 5430 . . 3 (𝑥 I 𝐴𝑥 = 𝐴)
86, 7bitr4i 267 . 2 (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
91, 2, 3, 5, 8brtxpsd3 32330 1 (𝐴Singleton𝐵𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321   class class class wbr 4804   I cid 5173   × cxp 5264  Singletoncsingle 32272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-symdif 3987  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-eprel 5179  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7334  df-2nd 7335  df-txp 32288  df-singleton 32296
This theorem is referenced by:  elsingles  32352  fnsingle  32353  fvsingle  32354  brapply  32372  brsuccf  32375  funpartlem  32376
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