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Theorem brsingle 33382
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 33327 . 2 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 brxp 5604 . . 3 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
51, 2, 4mpbir2an 709 . 2 𝐴(V × V)𝐵
6 velsn 4586 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
71ideq 5726 . . 3 (𝑥 I 𝐴𝑥 = 𝐴)
86, 7bitr4i 280 . 2 (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
91, 2, 3, 5, 8brtxpsd3 33361 1 (𝐴Singleton𝐵𝐵 = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 208   = wceq 1536  wcel 2113  Vcvv 3497  {csn 4570   class class class wbr 5069   I cid 5462   × cxp 5556  Singletoncsingle 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-eprel 5468  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-1st 7692  df-2nd 7693  df-txp 33319  df-singleton 33327
This theorem is referenced by:  elsingles  33383  fnsingle  33384  fvsingle  33385  brapply  33403  brsuccf  33406  funpartlem  33407
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