Theorem funopsn 5761

Description: If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 5319, as relsnopg 4778 is to relop 4827. (New usage is discouraged.)

Hypotheses

Ref	Expression
funopsn.x	⊢ 𝑋 ∈ V
funopsn.y	⊢ 𝑌 ∈ V

Assertion

Ref	Expression
funopsn	⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Distinct variable groups:   𝐹,𝑎   𝑋,𝑎   𝑌,𝑎

Proof of Theorem funopsn

Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopsn.x	. . 3 ⊢ 𝑋 ∈ V
2		funopsn.y	. . 3 ⊢ 𝑌 ∈ V
3		opm 4277	. . 3 ⊢ (∃𝑏 𝑏 ∈ ⟨𝑋, 𝑌⟩ ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V))
4	1, 2, 3	mpbir2an 944	. 2 ⊢ ∃𝑏 𝑏 ∈ ⟨𝑋, 𝑌⟩
5		funrel 5287	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
6	5	ad2antrr 488	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → Rel 𝐹)
7		simpr 110	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → 𝑏 ∈ ⟨𝑋, 𝑌⟩)
8		simplr 528	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → 𝐹 = ⟨𝑋, 𝑌⟩)
9	7, 8	eleqtrrd 2284	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → 𝑏 ∈ 𝐹)
10		elrel 4776	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑏 ∈ 𝐹) → ∃𝑐∃𝑑 𝑏 = ⟨𝑐, 𝑑⟩)
11	6, 9, 10	syl2anc 411	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → ∃𝑐∃𝑑 𝑏 = ⟨𝑐, 𝑑⟩)
12		vex 2774	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
13		vex 2774	. . . . . . . . . . . . . . . 16 ⊢ 𝑑 ∈ V
14	12, 13	dfop 3817	. . . . . . . . . . . . . . 15 ⊢ ⟨𝑐, 𝑑⟩ = {{𝑐}, {𝑐, 𝑑}}
15	14	eqeq2i 2215	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ ↔ 𝑏 = {{𝑐}, {𝑐, 𝑑}})
16	15	biimpi 120	. . . . . . . . . . . . 13 ⊢ (𝑏 = ⟨𝑐, 𝑑⟩ → 𝑏 = {{𝑐}, {𝑐, 𝑑}})
17	16	adantl 277	. . . . . . . . . . . 12 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → 𝑏 = {{𝑐}, {𝑐, 𝑑}})
18		vex 2774	. . . . . . . . . . . . . . . . 17 ⊢ 𝑏 ∈ V
19	18, 1, 2	elop 4274	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ ⟨𝑋, 𝑌⟩ ↔ (𝑏 = {𝑋} ∨ 𝑏 = {𝑋, 𝑌}))
20	7, 19	sylib 122	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → (𝑏 = {𝑋} ∨ 𝑏 = {𝑋, 𝑌}))
21		dfsn2 3646	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑋} = {𝑋, 𝑋}
22		simpl 109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → Fun 𝐹)
23		simpr 110	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → 𝐹 = ⟨𝑋, 𝑌⟩)
24	23	funeqd 5292	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (Fun 𝐹 ↔ Fun ⟨𝑋, 𝑌⟩))
25	22, 24	mpbid 147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → Fun ⟨𝑋, 𝑌⟩)
26		funopg 5304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ Fun ⟨𝑋, 𝑌⟩) → 𝑋 = 𝑌)
27	1, 2, 25, 26	mp3an12i 1353	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → 𝑋 = 𝑌)
28	27	preq2d 3716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → {𝑋, 𝑋} = {𝑋, 𝑌})
29	21, 28	eqtrid 2249	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → {𝑋} = {𝑋, 𝑌})
30	29	eqeq2d 2216	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝑏 = {𝑋} ↔ 𝑏 = {𝑋, 𝑌}))
31	30	orbi2d 791	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ((𝑏 = {𝑋} ∨ 𝑏 = {𝑋}) ↔ (𝑏 = {𝑋} ∨ 𝑏 = {𝑋, 𝑌})))
32	31	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → ((𝑏 = {𝑋} ∨ 𝑏 = {𝑋}) ↔ (𝑏 = {𝑋} ∨ 𝑏 = {𝑋, 𝑌})))
33	20, 32	mpbird 167	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → (𝑏 = {𝑋} ∨ 𝑏 = {𝑋}))
34		oridm 758	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = {𝑋} ∨ 𝑏 = {𝑋}) ↔ 𝑏 = {𝑋})
35	33, 34	sylib 122	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → 𝑏 = {𝑋})
36	35	adantr 276	. . . . . . . . . . . 12 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → 𝑏 = {𝑋})
37	17, 36	eqtr3d 2239	. . . . . . . . . . 11 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {{𝑐}, {𝑐, 𝑑}} = {𝑋})
38	12	snex 4228	. . . . . . . . . . . 12 ⊢ {𝑐} ∈ V
39		zfpair2 4253	. . . . . . . . . . . 12 ⊢ {𝑐, 𝑑} ∈ V
40	38, 39, 1	preqsn 3815	. . . . . . . . . . 11 ⊢ ({{𝑐}, {𝑐, 𝑑}} = {𝑋} ↔ ({𝑐} = {𝑐, 𝑑} ∧ {𝑐, 𝑑} = 𝑋))
41	37, 40	sylib 122	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → ({𝑐} = {𝑐, 𝑑} ∧ {𝑐, 𝑑} = 𝑋))
42	41	simpld 112	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {𝑐} = {𝑐, 𝑑})
43	41	simprd 114	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {𝑐, 𝑑} = 𝑋)
44	42, 43	eqtr2d 2238	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → 𝑋 = {𝑐})
45	1, 2	dfop 3817	. . . . . . . . . . 11 ⊢ ⟨𝑋, 𝑌⟩ = {{𝑋}, {𝑋, 𝑌}}
46		dfsn2 3646	. . . . . . . . . . . 12 ⊢ {{𝑋}} = {{𝑋}, {𝑋}}
47	29	preq2d 3716	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → {{𝑋}, {𝑋}} = {{𝑋}, {𝑋, 𝑌}})
48	46, 47	eqtrid 2249	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → {{𝑋}} = {{𝑋}, {𝑋, 𝑌}})
49	45, 23, 48	3eqtr4a 2263	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → 𝐹 = {{𝑋}})
50	49	ad2antrr 488	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → 𝐹 = {{𝑋}})
51	44	sneqd 3645	. . . . . . . . . . 11 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {𝑋} = {{𝑐}})
52	51	sneqd 3645	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {{𝑋}} = {{{𝑐}}})
53	12	opid 3836	. . . . . . . . . . 11 ⊢ ⟨𝑐, 𝑐⟩ = {{𝑐}}
54	53	sneqi 3644	. . . . . . . . . 10 ⊢ {⟨𝑐, 𝑐⟩} = {{{𝑐}}}
55	52, 54	eqtr4di 2255	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → {{𝑋}} = {⟨𝑐, 𝑐⟩})
56	50, 55	eqtrd 2237	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → 𝐹 = {⟨𝑐, 𝑐⟩})
57		sneq 3643	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → {𝑎} = {𝑐})
58	57	eqeq2d 2216	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → (𝑋 = {𝑎} ↔ 𝑋 = {𝑐}))
59		id 19	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → 𝑎 = 𝑐)
60	59, 59	opeq12d 3826	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ⟨𝑎, 𝑎⟩ = ⟨𝑐, 𝑐⟩)
61	60	sneqd 3645	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → {⟨𝑎, 𝑎⟩} = {⟨𝑐, 𝑐⟩})
62	61	eqeq2d 2216	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → (𝐹 = {⟨𝑎, 𝑎⟩} ↔ 𝐹 = {⟨𝑐, 𝑐⟩}))
63	58, 62	anbi12d 473	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → ((𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) ↔ (𝑋 = {𝑐} ∧ 𝐹 = {⟨𝑐, 𝑐⟩})))
64	12, 63	spcev 2867	. . . . . . . 8 ⊢ ((𝑋 = {𝑐} ∧ 𝐹 = {⟨𝑐, 𝑐⟩}) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
65	44, 56, 64	syl2anc 411	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) ∧ 𝑏 = ⟨𝑐, 𝑑⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
66	65	ex 115	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → (𝑏 = ⟨𝑐, 𝑑⟩ → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
67	66	exlimdvv 1920	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → (∃𝑐∃𝑑 𝑏 = ⟨𝑐, 𝑑⟩ → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
68	11, 67	mpd 13	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝑏 ∈ ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
69	68	expcom 116	. . 3 ⊢ (𝑏 ∈ ⟨𝑋, 𝑌⟩ → ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
70	69	exlimiv 1620	. 2 ⊢ (∃𝑏 𝑏 ∈ ⟨𝑋, 𝑌⟩ → ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
71	4, 70	ax-mp 5	1 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Vcvv 2771  {csn 3632  {cpr 3633  ⟨cop 3635  Rel wrel 4679  Fun wfun 5264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-fun 5272

This theorem is referenced by:  funop  5762