Theorem funopsn 6904

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.) (Avoid depending on this detail.)

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 6903	. 2 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
2		eqeq1 2825	. . . . . . 7 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}))
3		eqcom 2828	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩)
4	2, 3	syl6bb 289	. . . . . 6 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩))
5	4	adantl 484	. . . . 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 5358	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ≠ ∅
9		neeq1 3078	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
10	9	eqcoms 2829	. . . . . . . . . 10 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11		funrel 6366	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → Rel 𝐹)
12		reldm0 5792	. . . . . . . . . . . . . 14 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
13	11, 12	syl 17	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
14	13	biimprd 250	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
15	14	necon3d 3037	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
16	15	com12 32	. . . . . . . . . 10 ⊢ (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
17	10, 16	syl6bi 255	. . . . . . . . 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 413	. . . . . . 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 6677	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
22	21, 6, 7	iunopeqop 5403	. . . . . 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 242	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
25	24	imp 409	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26		iuneq1 4927	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩})
27		vex 3497	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
29		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
30	28, 29	opeq12d 4804	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑎, (𝐹‘𝑎)⟩)
31	30	sneqd 4572	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
32	27, 31	iunxsn 5005	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩}
33	26, 32	syl6eq 2872	. . . . . . . 8 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
34	33	adantl 484	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
35	34	eqeq2d 2832	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
36		eqeq1 2825	. . . . . . . . . . 11 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
37	36	adantl 484	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
38		eqcom 2828	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ {⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39		fvex 6677	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑎) ∈ V
40	27, 39	snopeqop 5388	. . . . . . . . . . 11 ⊢ ({⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
41	38, 40	sylbb 221	. . . . . . . . . 10 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
42	37, 41	syl6bi 255	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})))
43	42	imp 409	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
44		simpr3 1192	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝑋 = {𝑎})
45		simp1 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → 𝑎 = (𝐹‘𝑎))
46	45	eqcomd 2827	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹‘𝑎) = 𝑎)
47	46	opeq2d 4803	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝑎, 𝑎⟩)
48	47	sneqd 4572	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝑎, 𝑎⟩})
49	48	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
50	49	biimpac 481	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
51	44, 50	jca 514	. . . . . . . . . . 11 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
52	51	ex 415	. . . . . . . . . 10 ⊢ (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
53	52	adantl 484	. . . . . . . . 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 454	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
57	35, 56	sylbid 242	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
58	57	impancom 454	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
59	58	eximdv 1914	. . 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 687	1 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290  {csn 4560  ⟨cop 4566  ∪ ciun 4911  dom cdm 5549  Rel wrel 5554  Fun wfun 6343  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357

This theorem is referenced by:  funop  6905  funop1  43476