Theorem funopsn 7146

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.) (Proof shortened by Eric Schmidt, 9-May-2026.) 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 6589, as relsnopg 5792 is to relop 5838. (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 7145	. . 3 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
2		eqeq1 2767	. . . . . . 7 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}))
3		eqcom 2770	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩)
4	2, 3	bitrdi 290	. . . . . 6 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩))
5		fvex 6896	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
6		funopsn.x	. . . . . . 7 ⊢ 𝑋 ∈ V
7		funopsn.y	. . . . . . 7 ⊢ 𝑌 ∈ V
8	5, 6, 7	iunopeqop 5506	. . . . . 6 ⊢ (∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎})
9	4, 8	biimtrdi 256	. . . . 5 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
10	9	imp 411	. . . 4 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
11		iuneq1 4974	. . . . . . . . . 10 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩})
12		vex 3459	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
13		id 23	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
14		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
15	13, 14	opeq12d 4847	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑎, (𝐹‘𝑎)⟩)
16	15	sneqd 4602	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
17	12, 16	iunxsn 5058	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩}
18	11, 17	eqtrdi 2814	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
19	18	eqeq2d 2774	. . . . . . . 8 ⊢ (dom 𝐹 = {𝑎} → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
20	19	adantl 486	. . . . . . 7 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
21		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
22		eqcom 2770	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ {⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩)
23		fvex 6896	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑎) ∈ V
24	12, 23	snopeqop 5491	. . . . . . . . . . 11 ⊢ ({⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
25	22, 24	sylbb 222	. . . . . . . . . 10 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
26	21, 25	biimtrdi 256	. . . . . . . . 9 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})))
27		simpr3 1215	. . . . . . . . . . 11 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝑋 = {𝑎})
28		simp1 1154	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → 𝑎 = (𝐹‘𝑎))
29	28	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹‘𝑎) = 𝑎)
30	29	opeq2d 4846	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝑎, 𝑎⟩)
31	30	sneqd 4602	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝑎, 𝑎⟩})
32	31	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
33	32	biimpac 483	. . . . . . . . . . 11 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
34	27, 33	jca 520	. . . . . . . . . 10 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
35	34	ex 417	. . . . . . . . 9 ⊢ (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
36	26, 35	sylcom 31	. . . . . . . 8 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
37	36	adantr 485	. . . . . . 7 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
38	20, 37	sylbid 243	. . . . . 6 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
39	38	impancom 456	. . . . 5 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
40	39	eximdv 1947	. . . 4 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (∃𝑎dom 𝐹 = {𝑎} → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
41	10, 40	mpd 16	. . 3 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
42	1, 41	sylan2 604	. 2 ⊢ ((𝐹 = ⟨𝑋, 𝑌⟩ ∧ Fun 𝐹) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
43	42	ancoms 463	1 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  {csn 4590  ⟨cop 4596  ∪ ciun 4957  dom cdm 5663  Fun wfun 6532  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is referenced by:  funop  7148  funop1  47997