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Theorem fnsingle 34960
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 4131 . . . . 5 ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2 df-rel 5683 . . . . 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 34903 . . . . 5 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
54releqi 5777 . . . 4 (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
63, 5mpbir 230 . . 3 Rel Singleton
7 vex 3478 . . . . . . 7 𝑥 ∈ V
8 vex 3478 . . . . . . 7 𝑦 ∈ V
97, 8brsingle 34958 . . . . . 6 (𝑥Singleton𝑦𝑦 = {𝑥})
10 vex 3478 . . . . . . 7 𝑧 ∈ V
117, 10brsingle 34958 . . . . . 6 (𝑥Singleton𝑧𝑧 = {𝑥})
12 eqtr3 2758 . . . . . 6 ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
139, 11, 12syl2anb 598 . . . . 5 ((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1413ax-gen 1797 . . . 4 𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1514gen2 1798 . . 3 𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
16 dffun2 6553 . . 3 (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 709 . 2 Fun Singleton
18 eqv 3483 . . 3 (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19 eqid 2732 . . . . . 6 {𝑥} = {𝑥}
20 vsnex 5429 . . . . . . 7 {𝑥} ∈ V
217, 20brsingle 34958 . . . . . 6 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
2219, 21mpbir 230 . . . . 5 𝑥Singleton{𝑥}
23 breq2 5152 . . . . . 6 (𝑦 = {𝑥} → (𝑥Singleton𝑦𝑥Singleton{𝑥}))
2420, 23spcev 3596 . . . . 5 (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
2522, 24ax-mp 5 . . . 4 𝑦 𝑥Singleton𝑦
267eldm 5900 . . . 4 (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
2725, 26mpbir 230 . . 3 𝑥 ∈ dom Singleton
2818, 27mpgbir 1801 . 2 dom Singleton = V
29 df-fn 6546 . 2 (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
3017, 28, 29mpbir2an 709 1 Singleton Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cdif 3945  wss 3948  csymdif 4241  {csn 4628   class class class wbr 5148   I cid 5573   E cep 5579   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681  Fun wfun 6537   Fn wfn 6538  ctxp 34871  Singletoncsingle 34879
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977  df-2nd 7978  df-txp 34895  df-singleton 34903
This theorem is referenced by:  fvsingle  34961
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