Theorem funopsnOLD 7127

Description: Obsolete version of funopsn 7126 as of 9-May-2026. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
funopsn.x	⊢ 𝑋 ∈ V
funopsn.y	⊢ 𝑌 ∈ V

Assertion

Ref	Expression
funopsnOLD	⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Distinct variable groups:   𝐹,𝑎   𝑋,𝑎   𝑌,𝑎

Proof of Theorem funopsnOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funiun 7125	. 2 ⊢ (Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩})
2		eqeq1 2765	. . . . . . 7 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}))
3		eqcom 2768	. . . . . . 7 ⊢ (⟨𝑋, 𝑌⟩ = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩)
4	2, 3	bitrdi 289	. . . . . 6 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ⟨𝑋, 𝑌⟩))
5	4	adantl 485	. . . . 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 5441	. . . . . . 7 ⊢ ⟨𝑋, 𝑌⟩ ≠ ∅
9		neeq1 3018	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
10	9	eqcoms 2769	. . . . . . . . . 10 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11		funrel 6534	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → Rel 𝐹)
12		reldm0 5902	. . . . . . . . . . . . . 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 2977	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
16	15	com12 32	. . . . . . . . . 10 ⊢ (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
17	10, 16	biimtrdi 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 414	. . . . . . 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 6876	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
22	21, 6, 7	iunopeqopOLD 5490	. . . . . 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 410	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26		iuneq1 4965	. . . . . . . . 9 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩})
27		vex 3457	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
28		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
29		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
30	28, 29	opeq12d 4838	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑎, (𝐹‘𝑎)⟩)
31	30	sneqd 4593	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
32	27, 31	iunxsn 5047	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩}
33	26, 32	eqtrdi 2812	. . . . . . . 8 ⊢ (dom 𝐹 = {𝑎} → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
34	33	adantl 485	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} = {⟨𝑎, (𝐹‘𝑎)⟩})
35	34	eqeq2d 2772	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}))
36		eqeq1 2765	. . . . . . . . . . 11 ⊢ (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
37	36	adantl 485	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩}))
38		eqcom 2768	. . . . . . . . . . 11 ⊢ (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ {⟨𝑎, (𝐹‘𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39		fvex 6876	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑎) ∈ V
40	27, 39	snopeqop 5474	. . . . . . . . . . 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	biimtrdi 255	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})))
43	42	imp 410	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩}) → (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}))
44		simpr3 1209	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝑋 = {𝑎})
45		simp1 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → 𝑎 = (𝐹‘𝑎))
46	45	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹‘𝑎) = 𝑎)
47	46	opeq2d 4837	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝑎, 𝑎⟩)
48	47	sneqd 4593	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → {⟨𝑎, (𝐹‘𝑎)⟩} = {⟨𝑎, 𝑎⟩})
49	48	eqeq2d 2772	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
50	49	biimpac 482	. . . . . . . . . . . 12 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
51	44, 50	jca 519	. . . . . . . . . . 11 ⊢ ((𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} ∧ (𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
52	51	ex 416	. . . . . . . . . 10 ⊢ (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩} → ((𝑎 = (𝐹‘𝑎) ∧ 𝑋 = 𝑌 ∧ 𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
53	52	adantl 485	. . . . . . . . 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 455	. . . . . 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 455	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = ∪ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹‘𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
59	58	eximdv 1936	. . 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 699	1 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  ∅c0 4285  {csn 4581  ⟨cop 4587  ∪ ciun 4948  dom cdm 5645  Rel wrel 5650  Fun wfun 6511  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525

This theorem is referenced by: (None)