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Theorem fnsingle 32357
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 3886 . . . . 5 ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2 df-rel 5256 . . . . 5 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
31, 2mpbir 221 . . . 4 Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 df-singleton 32300 . . . . 5 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
54releqi 5342 . . . 4 (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
63, 5mpbir 221 . . 3 Rel Singleton
7 vex 3352 . . . . . . 7 𝑥 ∈ V
8 vex 3352 . . . . . . 7 𝑦 ∈ V
97, 8brsingle 32355 . . . . . 6 (𝑥Singleton𝑦𝑦 = {𝑥})
10 vex 3352 . . . . . . 7 𝑧 ∈ V
117, 10brsingle 32355 . . . . . 6 (𝑥Singleton𝑧𝑧 = {𝑥})
12 eqtr3 2791 . . . . . 6 ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
139, 11, 12syl2anb 577 . . . . 5 ((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1413ax-gen 1869 . . . 4 𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1514gen2 1870 . . 3 𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
16 dffun2 6041 . . 3 (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 682 . 2 Fun Singleton
18 eqv 3354 . . 3 (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19 eqid 2770 . . . . . 6 {𝑥} = {𝑥}
20 snex 5036 . . . . . . 7 {𝑥} ∈ V
217, 20brsingle 32355 . . . . . 6 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
2219, 21mpbir 221 . . . . 5 𝑥Singleton{𝑥}
23 breq2 4788 . . . . . 6 (𝑦 = {𝑥} → (𝑥Singleton𝑦𝑥Singleton{𝑥}))
2420, 23spcev 3449 . . . . 5 (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
2522, 24ax-mp 5 . . . 4 𝑦 𝑥Singleton𝑦
267eldm 5459 . . . 4 (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
2725, 26mpbir 221 . . 3 𝑥 ∈ dom Singleton
2818, 27mpgbir 1873 . 2 dom Singleton = V
29 df-fn 6034 . 2 (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
3017, 28, 29mpbir2an 682 1 Singleton Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1628   = wceq 1630  wex 1851  wcel 2144  Vcvv 3349  cdif 3718  wss 3721  csymdif 3990  {csn 4314   class class class wbr 4784   I cid 5156   E cep 5161   × cxp 5247  dom cdm 5249  ran crn 5250  Rel wrel 5254  Fun wfun 6025   Fn wfn 6026  ctxp 32268  Singletoncsingle 32276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-symdif 3991  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7314  df-2nd 7315  df-txp 32292  df-singleton 32300
This theorem is referenced by:  fvsingle  32358
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