Theorem funopg 6575

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 6592, as relsnopg 5796 is to relop 5843. (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 4868	. . . . 5 ⊢ (𝑢 = 𝐴 → ⟨𝑢, 𝑡⟩ = ⟨𝐴, 𝑡⟩)
2	1	funeqd 6563	. . . 4 ⊢ (𝑢 = 𝐴 → (Fun ⟨𝑢, 𝑡⟩ ↔ Fun ⟨𝐴, 𝑡⟩))
3		eqeq1 2730	. . . 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 4869	. . . . 5 ⊢ (𝑡 = 𝐵 → ⟨𝐴, 𝑡⟩ = ⟨𝐴, 𝐵⟩)
6	5	funeqd 6563	. . . 4 ⊢ (𝑡 = 𝐵 → (Fun ⟨𝐴, 𝑡⟩ ↔ Fun ⟨𝐴, 𝐵⟩))
7		eqeq2 2738	. . . 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 6558	. . . . 5 ⊢ (Fun ⟨𝑢, 𝑡⟩ → Rel ⟨𝑢, 𝑡⟩)
10		vex 3472	. . . . . 6 ⊢ 𝑢 ∈ V
11		vex 3472	. . . . . 6 ⊢ 𝑡 ∈ V
12	10, 11	relop 5843	. . . . 5 ⊢ (Rel ⟨𝑢, 𝑡⟩ ↔ ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
13	9, 12	sylib 217	. . . 4 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ∃𝑥∃𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
14	10, 11	opth 5469	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ (𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
15		vex 3472	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
16	15	opid 4888	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
17	16	preq1i 4735	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}} = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
18		vex 3472	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	15, 18	dfop 4867	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
20	19	preq2i 4736	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} = {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}}
21		vsnex 5422	. . . . . . . . . . 11 ⊢ {𝑥} ∈ V
22		zfpair2 5421	. . . . . . . . . . 11 ⊢ {𝑥, 𝑦} ∈ V
23	21, 22	dfop 4867	. . . . . . . . . 10 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
24	17, 20, 23	3eqtr4ri 2765	. . . . . . . . 9 ⊢ ⟨{𝑥}, {𝑥, 𝑦}⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
25	24	eqeq2i 2739	. . . . . . . 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 6552	. . . . . . . . 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 5457	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
30	29	prid1 4761	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
31		eleq2 2816	. . . . . . . . . 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 5457	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
34	33	prid2 4762	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
35		eleq2 2816	. . . . . . . . . 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 4870	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
39	38	3adant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
40	39	eleq1d 2812	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩))
41		opeq12 4870	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
42	41	3adant2 1128	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
43	42	eleq1d 2812	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
44	40, 43	anbi12d 630	. . . . . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑤 = 𝑣 ↔ 𝑥 = 𝑦))
46	45	3adant1 1127	. . . . . . . . . . 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 3588	. . . . . . . . 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 1457	. . . . . . . 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 241	. . . . . 6 ⊢ (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑥 = 𝑦))
52		dfsn2 4636	. . . . . . . . . . 11 ⊢ {𝑥} = {𝑥, 𝑥}
53		preq2 4733	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
54	52, 53	eqtr2id 2779	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
55	54	eqeq2d 2737	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} ↔ 𝑡 = {𝑥}))
56		eqtr3 2752	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥}) → 𝑢 = 𝑡)
57	56	expcom 413	. . . . . . . . 9 ⊢ (𝑡 = {𝑥} → (𝑢 = {𝑥} → 𝑢 = 𝑡))
58	55, 57	biimtrdi 252	. . . . . . . 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 1929	. . . 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 3557	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵))
65	64	3impia 1114	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3468  {csn 4623  {cpr 4625  ⟨cop 4629  Rel wrel 5674  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6538

This theorem is referenced by: (None)