Theorem funopg 6599

Description: A Kuratowski ordered pair of sets is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) 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 6616, as relsnopg 5812 is to relop 5860. (New usage is discouraged.)

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 4872	. . . . 5 ⊢ (𝑢 = 𝐴 → ⟨𝑢, 𝑡⟩ = ⟨𝐴, 𝑡⟩)
2	1	funeqd 6587	. . . 4 ⊢ (𝑢 = 𝐴 → (Fun ⟨𝑢, 𝑡⟩ ↔ Fun ⟨𝐴, 𝑡⟩))
3		eqeq1 2740	. . . 4 ⊢ (𝑢 = 𝐴 → (𝑢 = 𝑡 ↔ 𝐴 = 𝑡))
4	2, 3	imbi12d 344	. . 3 ⊢ (𝑢 = 𝐴 → ((Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡) ↔ (Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡)))
5		opeq2 4873	. . . . 5 ⊢ (𝑡 = 𝐵 → ⟨𝐴, 𝑡⟩ = ⟨𝐴, 𝐵⟩)
6	5	funeqd 6587	. . . 4 ⊢ (𝑡 = 𝐵 → (Fun ⟨𝐴, 𝑡⟩ ↔ Fun ⟨𝐴, 𝐵⟩))
7		eqeq2 2748	. . . 4 ⊢ (𝑡 = 𝐵 → (𝐴 = 𝑡 ↔ 𝐴 = 𝐵))
8	6, 7	imbi12d 344	. . 3 ⊢ (𝑡 = 𝐵 → ((Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡) ↔ (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵)))
9		funrel 6582	. . . . 5 ⊢ (Fun ⟨𝑢, 𝑡⟩ → Rel ⟨𝑢, 𝑡⟩)
10		vex 3483	. . . . . 6 ⊢ 𝑢 ∈ V
11		vex 3483	. . . . . 6 ⊢ 𝑡 ∈ V
12	10, 11	relop 5860	. . . . 5 ⊢ (Rel ⟨𝑢, 𝑡⟩ ↔ ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
13	9, 12	sylib 218	. . . 4 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
14	10, 11	opth 5480	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ (𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
15		vex 3483	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
16	15	opid 4892	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
17	16	preq1i 4735	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}} = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
18		vex 3483	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	15, 18	dfop 4871	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
20	19	preq2i 4736	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} = {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}}
21		vsnex 5433	. . . . . . . . . . 11 ⊢ {𝑥} ∈ V
22		zfpair2 5432	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ∈ V
23	21, 22	dfop 4871	. . . . . . . . . 10 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
24	17, 20, 23	3eqtr4ri 2775	. . . . . . . . 9 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
25	24	eqeq2i 2749	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
26	14, 25	bitr3i 277	. . . . . . 7 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
27		dffun4 6576	. . . . . . . . 9 ⊢ (Fun ⟨𝑢, 𝑡⟩ ↔ (Rel ⟨𝑢, 𝑡⟩ ∧ ∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣)))
28	27	simprbi 496	. . . . . . . 8 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ∀𝑧∀𝑤∀𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣))
29		opex 5468	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
30	29	prid1 4761	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
31		eleq2 2829	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
32	30, 31	mpbiri 258	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩)
33		opex 5468	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
34	33	prid2 4762	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
35		eleq2 2829	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
36	34, 35	mpbiri 258	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)
37	32, 36	jca 511	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
38		opeq12 4874	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
39	38	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
40	39	eleq1d 2825	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩))
41		opeq12 4874	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
42	41	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
43	42	eleq1d 2825	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
44	40, 43	anbi12d 632	. . . . . . . . . . 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 2753	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑤 = 𝑣 ↔ 𝑥 = 𝑦))
46	45	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑤 = 𝑣 ↔ 𝑥 = 𝑦))
47	44, 46	imbi12d 344	. . . . . . . . . 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 3603	. . . . . . . . 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 1462	. . . . . . . 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 40	. . . . . . 7 ⊢ (Fun ⟨𝑢, 𝑡⟩ → (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → 𝑥 = 𝑦))
51	26, 50	biimtrid 242	. . . . . 6 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑥 = 𝑦))
52		dfsn2 4638	. . . . . . . . . . 11 ⊢ {𝑥} = {𝑥, 𝑥}
53		preq2 4733	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
54	52, 53	eqtr2id 2789	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
55	54	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} ↔ 𝑡 = {𝑥}))
56		eqtr3 2762	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥}) → 𝑢 = 𝑡)
57	56	expcom 413	. . . . . . . . 9 ⊢ (𝑡 = {𝑥} → (𝑢 = {𝑥} → 𝑢 = 𝑡))
58	55, 57	biimtrdi 253	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} → (𝑢 = {𝑥} → 𝑢 = 𝑡)))
59	58	com13 88	. . . . . . 7 ⊢ (𝑢 = {𝑥} → (𝑡 = {𝑥, 𝑦} → (𝑥 = 𝑦 → 𝑢 = 𝑡)))
60	59	imp 406	. . . . . 6 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → (𝑥 = 𝑦 → 𝑢 = 𝑡))
61	51, 60	sylcom 30	. . . . 5 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
62	61	exlimdvv 1933	. . . 4 ⊢ (Fun ⟨𝑢, 𝑡⟩ → (∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
63	13, 62	mpd 15	. . 3 ⊢ (Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡)
64	4, 8, 63	vtocl2g 3573	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵))
65	64	3impia 1117	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479  {csn 4625  {cpr 4627  ⟨cop 4631  Rel wrel 5689  Fun wfun 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-fun 6562

This theorem is referenced by: (None)