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Theorem brsingle 31693
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 31637 . 2 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 brxp 5112 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 954 . 2 𝐴(V × V)𝐵
6 velsn 4169 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
71ideq 5239 . . 3 (𝑥 I 𝐴𝑥 = 𝐴)
86, 7bitr4i 267 . 2 (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
91, 2, 3, 5, 8brtxpsd3 31672 1 (𝐴Singleton𝐵𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  Vcvv 3189  {csn 4153   class class class wbr 4618   I cid 4989   × cxp 5077  Singletoncsingle 31613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31629  df-singleton 31637
This theorem is referenced by:  elsingles  31694  fnsingle  31695  fvsingle  31696  brapply  31714  brsuccf  31717  funpartlem  31718
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