Theorem funopsn 7120

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 6567, as relsnopg 5766 is to relop 5814. (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 variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funiun 7119	. 2 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
2		eqeq1 2733	. . . . . . 7 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}))
3		eqcom 2736	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩)
4	2, 3	bitrdi 287	. . . . . 6 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩))
5	4	adantl 481	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩))
6		funopsn.x	. . . . . . . 8 ⊢ 𝑋 ∈ V
7		funopsn.y	. . . . . . . 8 ⊢ 𝑌 ∈ V
8	6, 7	opnzi 5434	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ≠ ∅
9		neeq1 2987	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
10	9	eqcoms 2737	. . . . . . . . . 10 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11		funrel 6533	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → Rel 𝐹)
12		reldm0 5891	. . . . . . . . . . . . . 14 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
13	11, 12	syl 17	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
14	13	biimprd 248	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
15	14	necon3d 2946	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
16	15	com12 32	. . . . . . . . . 10 ⊢ (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
17	10, 16	biimtrdi 253	. . . . . . . . 9 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅)))
18	17	com3l 89	. . . . . . . 8 ⊢ (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → dom 𝐹 ≠ ∅)))
19	18	impd 410	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ ≠ ∅ → ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅))
20	8, 19	ax-mp 5	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅)
21		fvex 6871	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
22	21, 6, 7	iunopeqop 5481	. . . . . 6 ⊢ (dom 𝐹 ≠ ∅ → (∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
23	20, 22	syl 17	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
24	5, 23	sylbid 240	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
25	24	imp 406	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26		iuneq1 4972	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩})
27		vex 3451	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
29		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
30	28, 29	opeq12d 4845	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑎, (𝐹‘𝑎)⟩)
31	30	sneqd 4601	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
32	27, 31	iunxsn 5055	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩}
33	26, 32	eqtrdi 2780	. . . . . . . 8 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
34	33	adantl 481	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
35	34	eqeq2d 2740	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
36		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
38		eqcom 2736	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ {⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39		fvex 6871	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑎) ∈ V
40	27, 39	snopeqop 5466	. . . . . . . . . . 11 ⊢ ({⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
41	38, 40	sylbb 219	. . . . . . . . . 10 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
42	37, 41	biimtrdi 253	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})))
43	42	imp 406	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
44		simpr3 1197	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝑋 = {𝑎})
45		simp1 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → 𝑎 = (𝐹‘𝑎))
46	45	eqcomd 2735	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹‘𝑎) = 𝑎)
47	46	opeq2d 4844	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝑎, 𝑎⟩)
48	47	sneqd 4601	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝑎, 𝑎⟩})
49	48	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
50	49	biimpac 478	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
51	44, 50	jca 511	. . . . . . . . . . 11 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
52	51	ex 412	. . . . . . . . . 10 ⊢ (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
53	52	adantl 481	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
54	53	a1dd 50	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))))
55	43, 54	mpd 15	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
56	55	impancom 451	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
57	35, 56	sylbid 240	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
58	57	impancom 451	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
59	58	eximdv 1917	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (∃𝑎dom 𝐹 = {𝑎} → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
60	25, 59	mpd 15	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
61	1, 60	mpidan 689	1 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ∅c0 4296  {csn 4589  ⟨cop 4595  ∪ ciun 4955  dom cdm 5638  Rel wrel 5643  Fun wfun 6505  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  funop  7121  funop1  47284