Theorem fvsingle 36116

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 6834	. . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2		sneq 4578	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	1, 2	eqeq12d 2753	. . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4		eqid 2737	. . . . 5 ⊢ {𝑥} = {𝑥}
5		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
6		vsnex 5372	. . . . . 6 ⊢ {𝑥} ∈ V
7	5, 6	brsingle 36113	. . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
8	4, 7	mpbir 231	. . . 4 ⊢ 𝑥Singleton{𝑥}
9		fnsingle 36115	. . . . 5 ⊢ Singleton Fn V
10		fnbrfvb 6884	. . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
11	9, 5, 10	mp2an 693	. . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
12	8, 11	mpbir 231	. . 3 ⊢ (Singleton‘𝑥) = {𝑥}
13	3, 12	vtoclg 3500	. 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14		fvprc 6826	. . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15		snprc 4662	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
16	15	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
17	14, 16	eqtr4d 2775	. 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
18	13, 17	pm2.61i 182	1 ⊢ (Singleton‘𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  {csn 4568   class class class wbr 5086   Fn wfn 6487  ‘cfv 6492  Singletoncsingle 36034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7935  df-2nd 7936  df-txp 36050  df-singleton 36058

This theorem is referenced by: (None)