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Theorem brsingle 31194
 Description: The binary relationship 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 31138 . 2 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 brxp 5071 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 957 . 2 𝐴(V × V)𝐵
6 velsn 4141 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
71ideq 5196 . . 3 (𝑥 I 𝐴𝑥 = 𝐴)
86, 7bitr4i 266 . 2 (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
91, 2, 3, 5, 8brtxpsd3 31173 1 (𝐴Singleton𝐵𝐵 = {𝐴})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   class class class wbr 4583   I cid 4948   × cxp 5036  Singletoncsingle 31114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138 This theorem is referenced by:  elsingles  31195  fnsingle  31196  fvsingle  31197  brapply  31215  brsuccf  31218  funpartlem  31219
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