Theorem funopg 5232

Description: A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
funopg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Proof of Theorem funopg

Dummy variables 𝑢 𝑡 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3765	. . . . 5 ⊢ (𝑢 = 𝐴 → ⟨𝑢, 𝑡⟩ = ⟨𝐴, 𝑡⟩)
2	1	funeqd 5220	. . . 4 ⊢ (𝑢 = 𝐴 → (Fun ⟨𝑢, 𝑡⟩ ↔ Fun ⟨𝐴, 𝑡⟩))
3		eqeq1 2177	. . . 4 ⊢ (𝑢 = 𝐴 → (𝑢 = 𝑡 ↔ 𝐴 = 𝑡))
4	2, 3	imbi12d 233	. . 3 ⊢ (𝑢 = 𝐴 → ((Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡) ↔ (Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡)))
5		opeq2 3766	. . . . 5 ⊢ (𝑡 = 𝐵 → ⟨𝐴, 𝑡⟩ = ⟨𝐴, 𝐵⟩)
6	5	funeqd 5220	. . . 4 ⊢ (𝑡 = 𝐵 → (Fun ⟨𝐴, 𝑡⟩ ↔ Fun ⟨𝐴, 𝐵⟩))
7		eqeq2 2180	. . . 4 ⊢ (𝑡 = 𝐵 → (𝐴 = 𝑡 ↔ 𝐴 = 𝐵))
8	6, 7	imbi12d 233	. . 3 ⊢ (𝑡 = 𝐵 → ((Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡) ↔ (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵)))
9		funrel 5215	. . . . 5 ⊢ (Fun ⟨𝑢, 𝑡⟩ → Rel ⟨𝑢, 𝑡⟩)
10		vex 2733	. . . . . 6 ⊢ 𝑢 ∈ V
11		vex 2733	. . . . . 6 ⊢ 𝑡 ∈ V
12	10, 11	relop 4761	. . . . 5 ⊢ (Rel ⟨𝑢, 𝑡⟩ ↔ ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
13	9, 12	sylib 121	. . . 4 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
14	10, 11	opth 4222	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ (𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
15		vex 2733	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
16	15	opid 3783	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
17	16	preq1i 3663	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}} = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
18		vex 2733	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	15, 18	dfop 3764	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
20	19	preq2i 3664	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} = {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}}
21	15	snex 4171	. . . . . . . . . . 11 ⊢ {𝑥} ∈ V
22		zfpair2 4195	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ∈ V
23	21, 22	dfop 3764	. . . . . . . . . 10 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
24	17, 20, 23	3eqtr4ri 2202	. . . . . . . . 9 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
25	24	eqeq2i 2181	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
26	14, 25	bitr3i 185	. . . . . . 7 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
27		dffun4 5209	. . . . . . . . 9 ⊢ (Fun ⟨𝑢, 𝑡⟩ ↔ (Rel ⟨𝑢, 𝑡⟩ ∧ ∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣)))
28	27	simprbi 273	. . . . . . . 8 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣))
29	15, 15	opex 4214	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
30	29	prid1 3689	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
31		eleq2 2234	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
32	30, 31	mpbiri 167	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩)
33	15, 18	opex 4214	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
34	33	prid2 3690	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
35		eleq2 2234	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
36	34, 35	mpbiri 167	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)
37	32, 36	jca 304	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
38		opeq12 3767	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
39	38	3adant3 1012	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
40	39	eleq1d 2239	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩))
41		opeq12 3767	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
42	41	3adant2 1011	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
43	42	eleq1d 2239	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
44	40, 43	anbi12d 470	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) ↔ (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)))
45		eqeq12 2183	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑤 = 𝑣 ↔ 𝑥 = 𝑦))
46	45	3adant1 1010	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑤 = 𝑣 ↔ 𝑥 = 𝑦))
47	44, 46	imbi12d 233	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) ↔ ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦)))
48	47	spc3gv 2823	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) → ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦)))
49	15, 15, 18, 48	mp3an 1332	. . . . . . . 8 ⊢ (∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) → ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦))
50	28, 37, 49	syl2im 38	. . . . . . 7 ⊢ (Fun ⟨𝑢, 𝑡⟩ → (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → 𝑥 = 𝑦))
51	26, 50	syl5bi 151	. . . . . 6 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑥 = 𝑦))
52		dfsn2 3597	. . . . . . . . . . 11 ⊢ {𝑥} = {𝑥, 𝑥}
53		preq2 3661	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
54	52, 53	eqtr2id 2216	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
55	54	eqeq2d 2182	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} ↔ 𝑡 = {𝑥}))
56		eqtr3 2190	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥}) → 𝑢 = 𝑡)
57	56	expcom 115	. . . . . . . . 9 ⊢ (𝑡 = {𝑥} → (𝑢 = {𝑥} → 𝑢 = 𝑡))
58	55, 57	syl6bi 162	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} → (𝑢 = {𝑥} → 𝑢 = 𝑡)))
59	58	com13 80	. . . . . . 7 ⊢ (𝑢 = {𝑥} → (𝑡 = {𝑥, 𝑦} → (𝑥 = 𝑦 → 𝑢 = 𝑡)))
60	59	imp 123	. . . . . 6 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → (𝑥 = 𝑦 → 𝑢 = 𝑡))
61	51, 60	sylcom 28	. . . . 5 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
62	61	exlimdvv 1890	. . . 4 ⊢ (Fun ⟨𝑢, 𝑡⟩ → (∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
63	13, 62	mpd 13	. . 3 ⊢ (Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡)
64	4, 8, 63	vtocl2g 2794	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵))
65	64	3impia 1195	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  {csn 3583  {cpr 3584  ⟨cop 3586  Rel wrel 4616  Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-fun 5200

This theorem is referenced by: (None)