Theorem fnsingle 36777

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 4071	. . . . 5 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2		df-rel 5658	. . . . 5 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V))
3	1, 2	mpbir 234	. . . 4 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
4		df-singleton 36720	. . . . 5 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
5	4	releqi 5755	. . . 4 ⊢ (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
6	3, 5	mpbir 234	. . 3 ⊢ Rel Singleton
7		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brsingle 36775	. . . . . 6 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
10		vex 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
11	7, 10	brsingle 36775	. . . . . 6 ⊢ (𝑥Singleton𝑧 ↔ 𝑧 = {𝑥})
12		eqtr3 2782	. . . . . 6 ⊢ ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 610	. . . . 5 ⊢ ((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
14	13	ax-gen 1827	. . . 4 ⊢ ∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
15	14	gen2 1828	. . 3 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
16		dffun2 6553	. . 3 ⊢ (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 724	. 2 ⊢ Fun Singleton
18		eqv 3460	. . 3 ⊢ (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19		eqid 2760	. . . . . 6 ⊢ {𝑥} = {𝑥}
20		vsnex 5393	. . . . . . 7 ⊢ {𝑥} ∈ V
21	7, 20	brsingle 36775	. . . . . 6 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥})
22	19, 21	mpbir 234	. . . . 5 ⊢ 𝑥Singleton{𝑥}
23		breq2 5106	. . . . . 6 ⊢ (𝑦 = {𝑥} → (𝑥Singleton𝑦 ↔ 𝑥Singleton{𝑥}))
24	20, 23	spcev 3560	. . . . 5 ⊢ (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
25	22, 24	ax-mp 5	. . . 4 ⊢ ∃𝑦 𝑥Singleton𝑦
26	7	eldm 5883	. . . 4 ⊢ (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
27	25, 26	mpbir 234	. . 3 ⊢ 𝑥 ∈ dom Singleton
28	18, 27	mpgbir 1831	. 2 ⊢ dom Singleton = V
29		df-fn 6546	. 2 ⊢ (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
30	17, 28, 29	mpbir2an 724	1 ⊢ Singleton Fn V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   △ csymdif 4196  {csn 4584   class class class wbr 5102   I cid 5545   E cep 5550   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656  Fun wfun 6537   Fn wfn 6538   ⊗ ctxp 36688  Singletoncsingle 36696

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-symdif 4197  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 8005  df-2nd 8006  df-txp 36712  df-singleton 36720

This theorem is used by:  fvsingle  36778