Theorem brsingle 34268

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	⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Proof of Theorem brsingle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsingle.1	. 2 ⊢ 𝐴 ∈ V
2		brsingle.2	. 2 ⊢ 𝐵 ∈ V
3		df-singleton 34213	. 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		brxp 5647	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 709	. 2 ⊢ 𝐴(V × V)𝐵
6		velsn 4581	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
7	1	ideq 5774	. . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴)
8	6, 7	bitr4i 278	. 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
9	1, 2, 3, 5, 8	brtxpsd3 34247	1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2104  Vcvv 3437  {csn 4565   class class class wbr 5081   I cid 5499   × cxp 5598  Singletoncsingle 34189

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-symdif 4182  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-eprel 5506  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fo 6464  df-fv 6466  df-1st 7863  df-2nd 7864  df-txp 34205  df-singleton 34213

This theorem is referenced by:  elsingles  34269  fnsingle  34270  fvsingle  34271  brapply  34289  brsuccf  34292  funpartlem  34293