Theorem fvsingle 34201

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 6768	. . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2		sneq 4576	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	1, 2	eqeq12d 2755	. . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4		eqid 2739	. . . . 5 ⊢ {𝑥} = {𝑥}
5		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
6		snex 5357	. . . . . 6 ⊢ {𝑥} ∈ V
7	5, 6	brsingle 34198	. . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
8	4, 7	mpbir 230	. . . 4 ⊢ 𝑥Singleton{𝑥}
9		fnsingle 34200	. . . . 5 ⊢ Singleton Fn V
10		fnbrfvb 6816	. . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
11	9, 5, 10	mp2an 688	. . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
12	8, 11	mpbir 230	. . 3 ⊢ (Singleton‘𝑥) = {𝑥}
13	3, 12	vtoclg 3503	. 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14		fvprc 6760	. . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15		snprc 4658	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 215	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	14, 16	eqtr4d 2782	. 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
18	13, 17	pm2.61i 182	1 ⊢ (Singleton‘𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ∅c0 4261  {csn 4566   class class class wbr 5078   Fn wfn 6425  ‘cfv 6430  Singletoncsingle 34119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-symdif 4181  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-eprel 5494  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-1st 7817  df-2nd 7818  df-txp 34135  df-singleton 34143

This theorem is referenced by: (None)