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Theorem fnsingle 34891
Description: The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fnsingle Singleton Fn V

Proof of Theorem fnsingle
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 4132 . . . . 5 ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2 df-rel 5684 . . . . 5 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
31, 2mpbir 230 . . . 4 Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 df-singleton 34834 . . . . 5 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
54releqi 5778 . . . 4 (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
63, 5mpbir 230 . . 3 Rel Singleton
7 vex 3479 . . . . . . 7 𝑥 ∈ V
8 vex 3479 . . . . . . 7 𝑦 ∈ V
97, 8brsingle 34889 . . . . . 6 (𝑥Singleton𝑦𝑦 = {𝑥})
10 vex 3479 . . . . . . 7 𝑧 ∈ V
117, 10brsingle 34889 . . . . . 6 (𝑥Singleton𝑧𝑧 = {𝑥})
12 eqtr3 2759 . . . . . 6 ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
139, 11, 12syl2anb 599 . . . . 5 ((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1413ax-gen 1798 . . . 4 𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1514gen2 1799 . . 3 𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
16 dffun2 6554 . . 3 (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 710 . 2 Fun Singleton
18 eqv 3484 . . 3 (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19 eqid 2733 . . . . . 6 {𝑥} = {𝑥}
20 vsnex 5430 . . . . . . 7 {𝑥} ∈ V
217, 20brsingle 34889 . . . . . 6 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
2219, 21mpbir 230 . . . . 5 𝑥Singleton{𝑥}
23 breq2 5153 . . . . . 6 (𝑦 = {𝑥} → (𝑥Singleton𝑦𝑥Singleton{𝑥}))
2420, 23spcev 3597 . . . . 5 (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
2522, 24ax-mp 5 . . . 4 𝑦 𝑥Singleton𝑦
267eldm 5901 . . . 4 (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
2725, 26mpbir 230 . . 3 𝑥 ∈ dom Singleton
2818, 27mpgbir 1802 . 2 dom Singleton = V
29 df-fn 6547 . 2 (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
3017, 28, 29mpbir2an 710 1 Singleton Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cdif 3946  wss 3949  csymdif 4242  {csn 4629   class class class wbr 5149   I cid 5574   E cep 5580   × cxp 5675  dom cdm 5677  ran crn 5678  Rel wrel 5682  Fun wfun 6538   Fn wfn 6539  ctxp 34802  Singletoncsingle 34810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-singleton 34834
This theorem is referenced by:  fvsingle  34892
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