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Theorem fvsingle 34271
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.
StepHypRef Expression
1 fveq2 6804 . . . 4 (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2 sneq 4575 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
31, 2eqeq12d 2752 . . 3 (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4 eqid 2736 . . . . 5 {𝑥} = {𝑥}
5 vex 3441 . . . . . 6 𝑥 ∈ V
6 snex 5363 . . . . . 6 {𝑥} ∈ V
75, 6brsingle 34268 . . . . 5 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
84, 7mpbir 230 . . . 4 𝑥Singleton{𝑥}
9 fnsingle 34270 . . . . 5 Singleton Fn V
10 fnbrfvb 6854 . . . . 5 ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
119, 5, 10mp2an 690 . . . 4 ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
128, 11mpbir 230 . . 3 (Singleton‘𝑥) = {𝑥}
133, 12vtoclg 3510 . 2 (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14 fvprc 6796 . . 3 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15 snprc 4657 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
1714, 16eqtr4d 2779 . 2 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
1813, 17pm2.61i 182 1 (Singleton‘𝐴) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1539  wcel 2104  Vcvv 3437  c0 4262  {csn 4565   class class class wbr 5081   Fn wfn 6453  cfv 6458  Singletoncsingle 34189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-symdif 4182  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-eprel 5506  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fo 6464  df-fv 6466  df-1st 7863  df-2nd 7864  df-txp 34205  df-singleton 34213
This theorem is referenced by: (None)
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