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Theorem fnsingle 31196
 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 3699 . . . . 5 ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2 df-rel 5045 . . . . 5 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
31, 2mpbir 220 . . . 4 Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 df-singleton 31138 . . . . 5 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
54releqi 5125 . . . 4 (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
63, 5mpbir 220 . . 3 Rel Singleton
7 vex 3176 . . . . . . 7 𝑥 ∈ V
8 vex 3176 . . . . . . 7 𝑦 ∈ V
97, 8brsingle 31194 . . . . . 6 (𝑥Singleton𝑦𝑦 = {𝑥})
10 vex 3176 . . . . . . 7 𝑧 ∈ V
117, 10brsingle 31194 . . . . . 6 (𝑥Singleton𝑧𝑧 = {𝑥})
12 eqtr3 2631 . . . . . 6 ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
139, 11, 12syl2anb 495 . . . . 5 ((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1413ax-gen 1713 . . . 4 𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1514gen2 1714 . . 3 𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
16 dffun2 5814 . . 3 (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 957 . 2 Fun Singleton
18 eqv 3178 . . 3 (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19 eqid 2610 . . . . . 6 {𝑥} = {𝑥}
20 snex 4835 . . . . . . 7 {𝑥} ∈ V
217, 20brsingle 31194 . . . . . 6 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
2219, 21mpbir 220 . . . . 5 𝑥Singleton{𝑥}
23 breq2 4587 . . . . . 6 (𝑦 = {𝑥} → (𝑥Singleton𝑦𝑥Singleton{𝑥}))
2420, 23spcev 3273 . . . . 5 (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
2522, 24ax-mp 5 . . . 4 𝑦 𝑥Singleton𝑦
267eldm 5243 . . . 4 (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
2725, 26mpbir 220 . . 3 𝑥 ∈ dom Singleton
2818, 27mpgbir 1717 . 2 dom Singleton = V
29 df-fn 5807 . 2 (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
3017, 28, 29mpbir2an 957 1 Singleton Fn V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   △ csymdif 3805  {csn 4125   class class class wbr 4583   E cep 4947   I cid 4948   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799   ⊗ ctxp 31106  Singletoncsingle 31114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138 This theorem is referenced by:  fvsingle  31197
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