Theorem funop1 47909

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 4844	. . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
2	1	eqeq2d 2780	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
3	2	cbvex2vw 2068	. 2 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4		vex 3467	. . . . . . 7 ⊢ 𝑣 ∈ V
5		vex 3467	. . . . . . 7 ⊢ 𝑤 ∈ V
6	4, 5	funopsn 7145	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7		vex 3467	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		opeq12 4844	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
9	8	sneqd 4606	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
10	9	eqeq2d 2780	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
11	7, 7, 10	spc2ev 3575	. . . . . . . 8 ⊢ (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
12	11	adantl 486	. . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
13	12	exlimiv 1957	. . . . . 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 418	. . . 4 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16		vex 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
17		vex 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
18	16, 17	funsn 6590	. . . . . 6 ⊢ Fun {⟨𝑥, 𝑦⟩}
19		funeq 6557	. . . . . 6 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
20	18, 19	mpbiri 261	. . . . 5 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
21	20	exlimivv 1959	. . . 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 1959	. 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 400   = wceq 1567  ∃wex 1806  {csn 4594  ⟨cop 4600  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by: (None)