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Theorem fnsingle 32539
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 3935 . . . . 5 ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2 df-rel 5319 . . . . 5 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
31, 2mpbir 223 . . . 4 Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4 df-singleton 32482 . . . . 5 Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
54releqi 5407 . . . 4 (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
63, 5mpbir 223 . . 3 Rel Singleton
7 vex 3388 . . . . . . 7 𝑥 ∈ V
8 vex 3388 . . . . . . 7 𝑦 ∈ V
97, 8brsingle 32537 . . . . . 6 (𝑥Singleton𝑦𝑦 = {𝑥})
10 vex 3388 . . . . . . 7 𝑧 ∈ V
117, 10brsingle 32537 . . . . . 6 (𝑥Singleton𝑧𝑧 = {𝑥})
12 eqtr3 2820 . . . . . 6 ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
139, 11, 12syl2anb 592 . . . . 5 ((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1413ax-gen 1891 . . . 4 𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
1514gen2 1892 . . 3 𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)
16 dffun2 6111 . . 3 (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥𝑦𝑧((𝑥Singleton𝑦𝑥Singleton𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 703 . 2 Fun Singleton
18 eqv 3391 . . 3 (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19 eqid 2799 . . . . . 6 {𝑥} = {𝑥}
20 snex 5099 . . . . . . 7 {𝑥} ∈ V
217, 20brsingle 32537 . . . . . 6 (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
2219, 21mpbir 223 . . . . 5 𝑥Singleton{𝑥}
23 breq2 4847 . . . . . 6 (𝑦 = {𝑥} → (𝑥Singleton𝑦𝑥Singleton{𝑥}))
2420, 23spcev 3488 . . . . 5 (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
2522, 24ax-mp 5 . . . 4 𝑦 𝑥Singleton𝑦
267eldm 5524 . . . 4 (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
2725, 26mpbir 223 . . 3 𝑥 ∈ dom Singleton
2818, 27mpgbir 1895 . 2 dom Singleton = V
29 df-fn 6104 . 2 (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
3017, 28, 29mpbir2an 703 1 Singleton Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  cdif 3766  wss 3769  csymdif 4040  {csn 4368   class class class wbr 4843   I cid 5219   E cep 5224   × cxp 5310  dom cdm 5312  ran crn 5313  Rel wrel 5317  Fun wfun 6095   Fn wfn 6096  ctxp 32450  Singletoncsingle 32458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-symdif 4041  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-eprel 5225  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-txp 32474  df-singleton 32482
This theorem is referenced by:  fvsingle  32540
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