Theorem brsingle 36273

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 36218	. 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		brxp 5700	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 723	. 2 ⊢ 𝐴(V × V)𝐵
6		velsn 4601	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
7	1	ideq 5828	. . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴)
8	6, 7	bitr4i 281	. 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
9	1, 2, 3, 5, 8	brtxpsd3 36252	1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  Vcvv 3457  {csn 4585   class class class wbr 5104   I cid 5545   × cxp 5649  Singletoncsingle 36194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-symdif 4208  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-1st 7974  df-2nd 7975  df-txp 36210  df-singleton 36218

This theorem is referenced by:  elsingles  36274  fnsingle  36275  fvsingle  36276  brapply  36294  lemsuccf  36297  funpartlem  36300