Theorem brsingle 34889

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 34834	. 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		brxp 5726	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 710	. 2 ⊢ 𝐴(V × V)𝐵
6		velsn 4645	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
7	1	ideq 5853	. . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴)
8	6, 7	bitr4i 278	. 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
9	1, 2, 3, 5, 8	brtxpsd3 34868	1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4629   class class class wbr 5149   I cid 5574   × cxp 5675  Singletoncsingle 34810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-singleton 34834

This theorem is referenced by:  elsingles  34890  fnsingle  34891  fvsingle  34892  brapply  34910  brsuccf  34913  funpartlem  34914