Theorem brsingle 36388

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 36333	. 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		brxp 5712	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 723	. 2 ⊢ 𝐴(V × V)𝐵
6		velsn 4606	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
7	1	ideq 5840	. . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴)
8	6, 7	bitr4i 281	. 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴)
9	1, 2, 3, 5, 8	brtxpsd3 36367	1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴})

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4590   class class class wbr 5110   I cid 5557   × cxp 5661  Singletoncsingle 36309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-symdif 4207  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 7987  df-2nd 7988  df-txp 36325  df-singleton 36333

This theorem is referenced by:  elsingles  36389  fnsingle  36390  fvsingle  36391  brapply  36409  lemsuccf  36412  funpartlem  36415