Theorem fnsingle 35953

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 4081	. . . . 5 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2		df-rel 5618	. . . . 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 35896	. . . . 5 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
5	4	releqi 5713	. . . 4 ⊢ (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
6	3, 5	mpbir 231	. . 3 ⊢ Rel Singleton
7		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brsingle 35951	. . . . . 6 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
10		vex 3440	. . . . . . 7 ⊢ 𝑧 ∈ V
11	7, 10	brsingle 35951	. . . . . 6 ⊢ (𝑥Singleton𝑧 ↔ 𝑧 = {𝑥})
12		eqtr3 2753	. . . . . 6 ⊢ ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 598	. . . . 5 ⊢ ((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
14	13	ax-gen 1796	. . . 4 ⊢ ∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
15	14	gen2 1797	. . 3 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
16		dffun2 6486	. . 3 ⊢ (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 711	. 2 ⊢ Fun Singleton
18		eqv 3446	. . 3 ⊢ (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19		eqid 2731	. . . . . 6 ⊢ {𝑥} = {𝑥}
20		vsnex 5367	. . . . . . 7 ⊢ {𝑥} ∈ V
21	7, 20	brsingle 35951	. . . . . 6 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
22	19, 21	mpbir 231	. . . . 5 ⊢ 𝑥Singleton{𝑥}
23		breq2 5090	. . . . . 6 ⊢ (𝑦 = {𝑥} → (𝑥Singleton𝑦 ↔ 𝑥Singleton{𝑥}))
24	20, 23	spcev 3556	. . . . 5 ⊢ (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
25	22, 24	ax-mp 5	. . . 4 ⊢ ∃𝑦 𝑥Singleton𝑦
26	7	eldm 5835	. . . 4 ⊢ (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
27	25, 26	mpbir 231	. . 3 ⊢ 𝑥 ∈ dom Singleton
28	18, 27	mpgbir 1800	. 2 ⊢ dom Singleton = V
29		df-fn 6479	. 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897   △ csymdif 4197  {csn 4571   class class class wbr 5086   I cid 5505   E cep 5510   × cxp 5609  dom cdm 5611  ran crn 5612  Rel wrel 5616  Fun wfun 6470   Fn wfn 6471   ⊗ ctxp 35864  Singletoncsingle 35872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-symdif 4198  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-eprel 5511  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fo 6482  df-fv 6484  df-1st 7916  df-2nd 7917  df-txp 35888  df-singleton 35896

This theorem is referenced by:  fvsingle  35954