Theorem fvsingle 33376

Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)

Assertion

Ref	Expression
fvsingle	⊢ (Singleton‘𝐴) = {𝐴}

Proof of Theorem fvsingle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6664	. . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2		sneq 4570	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	1, 2	eqeq12d 2837	. . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4		eqid 2821	. . . . 5 ⊢ {𝑥} = {𝑥}
5		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
6		snex 5323	. . . . . 6 ⊢ {𝑥} ∈ V
7	5, 6	brsingle 33373	. . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
8	4, 7	mpbir 233	. . . 4 ⊢ 𝑥Singleton{𝑥}
9		fnsingle 33375	. . . . 5 ⊢ Singleton Fn V
10		fnbrfvb 6712	. . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
11	9, 5, 10	mp2an 690	. . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
12	8, 11	mpbir 233	. . 3 ⊢ (Singleton‘𝑥) = {𝑥}
13	3, 12	vtoclg 3567	. 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14		fvprc 6657	. . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15		snprc 4646	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	14, 16	eqtr4d 2859	. 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
18	13, 17	pm2.61i 184	1 ⊢ (Singleton‘𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4560   class class class wbr 5058   Fn wfn 6344  ‘cfv 6349  Singletoncsingle 33294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-symdif 4218  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-eprel 5459  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-1st 7683  df-2nd 7684  df-txp 33310  df-singleton 33318

This theorem is referenced by: (None)