Theorem funop1 48626

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 4847	. . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
2	1	eqeq2d 2774	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
3	2	cbvex2vw 2076	. 2 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4		vex 3458	. . . . . . 7 ⊢ 𝑣 ∈ V
5		vex 3458	. . . . . . 7 ⊢ 𝑤 ∈ V
6	4, 5	funopsn 7183	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7		vex 3458	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		opeq12 4847	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
9	8	sneqd 4596	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
10	9	eqeq2d 2774	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
11	7, 7, 10	spc2ev 3565	. . . . . . . 8 ⊢ (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
12	11	adantl 487	. . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
13	12	exlimiv 1965	. . . . . 6 ⊢ (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
14	6, 13	syl 18	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
15	14	expcom 419	. . . 4 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
17		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
18	16, 17	funsn 6623	. . . . . 6 ⊢ Fun {⟨𝑥, 𝑦⟩}
19		funeq 6587	. . . . . 6 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
20	18, 19	mpbiri 261	. . . . 5 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
21	20	exlimivv 1967	. . . 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 1967	. 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⟩}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  {csn 4588  ⟨cop 4840  Fun wfun 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by: (None)