Theorem fnsingle 35883

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.

Step	Hyp	Ref	Expression
1		difss 4159	. . . . 5 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2		df-rel 5707	. . . . 5 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
3	1, 2	mpbir 231	. . . 4 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		df-singleton 35826	. . . . 5 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
5	4	releqi 5801	. . . 4 ⊢ (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
6	3, 5	mpbir 231	. . 3 ⊢ Rel Singleton
7		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brsingle 35881	. . . . . 6 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
10		vex 3492	. . . . . . 7 ⊢ 𝑧 ∈ V
11	7, 10	brsingle 35881	. . . . . 6 ⊢ (𝑥Singleton𝑧 ↔ 𝑧 = {𝑥})
12		eqtr3 2766	. . . . . 6 ⊢ ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 597	. . . . 5 ⊢ ((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
14	13	ax-gen 1793	. . . 4 ⊢ ∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
15	14	gen2 1794	. . 3 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
16		dffun2 6583	. . 3 ⊢ (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 710	. 2 ⊢ Fun Singleton
18		eqv 3498	. . 3 ⊢ (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19		eqid 2740	. . . . . 6 ⊢ {𝑥} = {𝑥}
20		vsnex 5449	. . . . . . 7 ⊢ {𝑥} ∈ V
21	7, 20	brsingle 35881	. . . . . 6 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
22	19, 21	mpbir 231	. . . . 5 ⊢ 𝑥Singleton{𝑥}
23		breq2 5170	. . . . . 6 ⊢ (𝑦 = {𝑥} → (𝑥Singleton𝑦 ↔ 𝑥Singleton{𝑥}))
24	20, 23	spcev 3619	. . . . 5 ⊢ (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
25	22, 24	ax-mp 5	. . . 4 ⊢ ∃𝑦 𝑥Singleton𝑦
26	7	eldm 5925	. . . 4 ⊢ (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
27	25, 26	mpbir 231	. . 3 ⊢ 𝑥 ∈ dom Singleton
28	18, 27	mpgbir 1797	. 2 ⊢ dom Singleton = V
29		df-fn 6576	. 2 ⊢ (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
30	17, 28, 29	mpbir2an 710	1 ⊢ Singleton Fn V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976   △ csymdif 4271  {csn 4648   class class class wbr 5166   I cid 5592   E cep 5598   × cxp 5698  dom cdm 5700  ran crn 5701  Rel wrel 5705  Fun wfun 6567   Fn wfn 6568   ⊗ ctxp 35794  Singletoncsingle 35802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-symdif 4272  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-1st 8030  df-2nd 8031  df-txp 35818  df-singleton 35826

This theorem is referenced by:  fvsingle  35884