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Theorem fvsingle 32471
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 6375 . . . 4 (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴))
2 sneq 4344 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
31, 2eqeq12d 2780 . . 3 (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴}))
4 eqid 2765 . . . . 5 {𝑥} = {𝑥}
5 vex 3353 . . . . . 6 𝑥 ∈ V
6 snex 5064 . . . . . 6 {𝑥} ∈ V
75, 6brsingle 32468 . . . . 5 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
84, 7mpbir 222 . . . 4 𝑥Singleton{𝑥}
9 fnsingle 32470 . . . . 5 Singleton Fn V
10 fnbrfvb 6424 . . . . 5 ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}))
119, 5, 10mp2an 683 . . . 4 ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})
128, 11mpbir 222 . . 3 (Singleton‘𝑥) = {𝑥}
133, 12vtoclg 3418 . 2 (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
14 fvprc 6368 . . 3 𝐴 ∈ V → (Singleton‘𝐴) = ∅)
15 snprc 4408 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
1615biimpi 207 . . 3 𝐴 ∈ V → {𝐴} = ∅)
1714, 16eqtr4d 2802 . 2 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴})
1813, 17pm2.61i 176 1 (Singleton‘𝐴) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  {csn 4334   class class class wbr 4809   Fn wfn 6063  cfv 6068  Singletoncsingle 32389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-symdif 4005  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-eprel 5190  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-txp 32405  df-singleton 32413
This theorem is referenced by: (None)
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