Theorem funop1 47198

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 4899	. . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑤⟩)
2	1	eqeq2d 2751	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = ⟨𝑥, 𝑦⟩ ↔ 𝐹 = ⟨𝑣, 𝑤⟩))
3	2	cbvex2vw 2040	. 2 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩)
4		vex 3492	. . . . . . 7 ⊢ 𝑣 ∈ V
5		vex 3492	. . . . . . 7 ⊢ 𝑤 ∈ V
6	4, 5	funopsn 7182	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑣, 𝑤⟩) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
7		vex 3492	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		opeq12 4899	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑎⟩)
9	8	sneqd 4660	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {⟨𝑥, 𝑦⟩} = {⟨𝑎, 𝑎⟩})
10	9	eqeq2d 2751	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {⟨𝑥, 𝑦⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
11	7, 7, 10	spc2ev 3620	. . . . . . . 8 ⊢ (𝐹 = {⟨𝑎, 𝑎⟩} → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
12	11	adantl 481	. . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}) → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩})
13	12	exlimiv 1929	. . . . . 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 413	. . . 4 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
16		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
17		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
18	16, 17	funsn 6631	. . . . . 6 ⊢ Fun {⟨𝑥, 𝑦⟩}
19		funeq 6598	. . . . . 6 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩}))
20	18, 19	mpbiri 258	. . . . 5 ⊢ (𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
21	20	exlimivv 1931	. . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩} → Fun 𝐹)
22	15, 21	impbid1 225	. . 3 ⊢ (𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
23	22	exlimivv 1931	. 2 ⊢ (∃𝑣∃𝑤 𝐹 = ⟨𝑣, 𝑤⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
24	3, 23	sylbi 217	1 ⊢ (∃𝑥∃𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777  {csn 4648  ⟨cop 4654  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by: (None)