Theorem fnsingle 36118

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 4077	. . . . 5 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) ⊆ (V × V)
2		df-rel 5632	. . . . 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 36061	. . . . 5 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
5	4	releqi 5728	. . . 4 ⊢ (Rel Singleton ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))))
6	3, 5	mpbir 231	. . 3 ⊢ Rel Singleton
7		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brsingle 36116	. . . . . 6 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
10		vex 3434	. . . . . . 7 ⊢ 𝑧 ∈ V
11	7, 10	brsingle 36116	. . . . . 6 ⊢ (𝑥Singleton𝑧 ↔ 𝑧 = {𝑥})
12		eqtr3 2759	. . . . . 6 ⊢ ((𝑦 = {𝑥} ∧ 𝑧 = {𝑥}) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 599	. . . . 5 ⊢ ((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
14	13	ax-gen 1797	. . . 4 ⊢ ∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
15	14	gen2 1798	. . 3 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)
16		dffun2 6503	. . 3 ⊢ (Fun Singleton ↔ (Rel Singleton ∧ ∀𝑥∀𝑦∀𝑧((𝑥Singleton𝑦 ∧ 𝑥Singleton𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 712	. 2 ⊢ Fun Singleton
18		eqv 3440	. . 3 ⊢ (dom Singleton = V ↔ ∀𝑥 𝑥 ∈ dom Singleton)
19		eqid 2737	. . . . . 6 ⊢ {𝑥} = {𝑥}
20		vsnex 5373	. . . . . . 7 ⊢ {𝑥} ∈ V
21	7, 20	brsingle 36116	. . . . . 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 3549	. . . . 5 ⊢ (𝑥Singleton{𝑥} → ∃𝑦 𝑥Singleton𝑦)
25	22, 24	ax-mp 5	. . . 4 ⊢ ∃𝑦 𝑥Singleton𝑦
26	7	eldm 5850	. . . 4 ⊢ (𝑥 ∈ dom Singleton ↔ ∃𝑦 𝑥Singleton𝑦)
27	25, 26	mpbir 231	. . 3 ⊢ 𝑥 ∈ dom Singleton
28	18, 27	mpgbir 1801	. 2 ⊢ dom Singleton = V
29		df-fn 6496	. 2 ⊢ (Singleton Fn V ↔ (Fun Singleton ∧ dom Singleton = V))
30	17, 28, 29	mpbir2an 712	1 ⊢ Singleton Fn V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890   △ csymdif 4193  {csn 4568   class class class wbr 5086   I cid 5519   E cep 5524   × cxp 5623  dom cdm 5625  ran crn 5626  Rel wrel 5630  Fun wfun 6487   Fn wfn 6488   ⊗ ctxp 36029  Singletoncsingle 36037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7936  df-2nd 7937  df-txp 36053  df-singleton 36061

This theorem is referenced by:  fvsingle  36119