Theorem fnsingle 35921

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 4135	. . . . 5 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2		df-rel 5691	. . . . 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 35864	. . . . 5 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
5	4	releqi 5786	. . . 4 ⊢ (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
6	3, 5	mpbir 231	. . 3 ⊢ Rel Singleton
7		vex 3483	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3483	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brsingle 35919	. . . . . 6 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
10		vex 3483	. . . . . . 7 ⊢ 𝑧 ∈ V
11	7, 10	brsingle 35919	. . . . . 6 ⊢ (𝑥Singleton𝑧 ↔ 𝑧 = {𝑥})
12		eqtr3 2762	. . . . . 6 ⊢ ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 598	. . . . 5 ⊢ ((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
14	13	ax-gen 1794	. . . 4 ⊢ ∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
15	14	gen2 1795	. . 3 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
16		dffun2 6570	. . 3 ⊢ (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 711	. 2 ⊢ Fun Singleton
18		eqv 3489	. . 3 ⊢ (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19		eqid 2736	. . . . . 6 ⊢ {𝑥} = {𝑥}
20		vsnex 5433	. . . . . . 7 ⊢ {𝑥} ∈ V
21	7, 20	brsingle 35919	. . . . . 6 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
22	19, 21	mpbir 231	. . . . 5 ⊢ 𝑥Singleton{𝑥}
23		breq2 5146	. . . . . 6 ⊢ (𝑦 = {𝑥} → (𝑥Singleton𝑦 ↔ 𝑥Singleton{𝑥}))
24	20, 23	spcev 3605	. . . . 5 ⊢ (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
25	22, 24	ax-mp 5	. . . 4 ⊢ ∃𝑦 𝑥Singleton𝑦
26	7	eldm 5910	. . . 4 ⊢ (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
27	25, 26	mpbir 231	. . 3 ⊢ 𝑥 ∈ dom Singleton
28	18, 27	mpgbir 1798	. 2 ⊢ dom Singleton = V
29		df-fn 6563	. 2 ⊢ (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
30	17, 28, 29	mpbir2an 711	1 ⊢ Singleton Fn V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479   ∖ cdif 3947   ⊆ wss 3950   △ csymdif 4251  {csn 4625   class class class wbr 5142   I cid 5576   E cep 5582   × cxp 5682  dom cdm 5684  ran crn 5685  Rel wrel 5689  Fun wfun 6554   Fn wfn 6555   ⊗ ctxp 35832  Singletoncsingle 35840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-symdif 4252  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-eprel 5583  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-txp 35856  df-singleton 35864

This theorem is referenced by:  fvsingle  35922