Theorem funop1 43839

Description: A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.)

Assertion

Ref	Expression
funop1	⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem funop1

Dummy variables 𝑎 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq12 4767	. . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
2	1	eqeq2d 2809	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
3	2	cbvex2vw 2048	. 2 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4		vex 3444	. . . . . . 7 ⊢ 𝑣 ∈ V
5		vex 3444	. . . . . . 7 ⊢ 𝑤 ∈ V
6	4, 5	funopsn 6887	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7		vex 3444	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		opeq12 4767	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
9	8	sneqd 4537	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
10	9	eqeq2d 2809	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
11	7, 7, 10	spc2ev 3556	. . . . . . . 8 ⊢ (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
12	11	adantl 485	. . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
13	12	exlimiv 1931	. . . . . 6 ⊢ (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
14	6, 13	syl 17	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
15	14	expcom 417	. . . 4 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
17		vex 3444	. . . . . . 7 ⊢ 𝑦 ∈ V
18	16, 17	funsn 6377	. . . . . 6 ⊢ Fun {⟨𝑥, 𝑦⟩}
19		funeq 6344	. . . . . 6 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
20	18, 19	mpbiri 261	. . . . 5 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
21	20	exlimivv 1933	. . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
22	15, 21	impbid1 228	. . 3 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
23	22	exlimivv 1933	. 2 ⊢ (∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
24	3, 23	sylbi 220	1 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  {csn 4525  ⟨cop 4531  Fun wfun 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by: (None)