Theorem dffun10 34216

Description: Another potential definition of functionality. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)

Assertion

Ref	Expression
dffun10	⊢ (Fun 𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))

Proof of Theorem dffun10

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 451	. . . . . . 7 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
2	1	albii 1822	. . . . . 6 ⊢ (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
3		19.21v 1942	. . . . . 6 ⊢ (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
4		vex 3436	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
5		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6	4, 5	opelco 5780	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦))
7		df-br 5075	. . . . . . . . . . . 12 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
8		brv 5387	. . . . . . . . . . . . . 14 ⊢ 𝑧V𝑦
9		brdif 5127	. . . . . . . . . . . . . 14 ⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
10	8, 9	mpbiran 706	. . . . . . . . . . . . 13 ⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
11	5	ideq 5761	. . . . . . . . . . . . . 14 ⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦)
12		equcom 2021	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
13	11, 12	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑧 I 𝑦 ↔ 𝑦 = 𝑧)
14	10, 13	xchbinx 334	. . . . . . . . . . . 12 ⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
15	7, 14	anbi12i 627	. . . . . . . . . . 11 ⊢ ((𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
16	15	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
17		exanali 1862	. . . . . . . . . 10 ⊢ (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))
18	6, 16, 17	3bitri 297	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))
19	18	con2bii 358	. . . . . . . 8 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
20		opex 5379	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
21		eldif 3897	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
22	20, 21	mpbiran 706	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
23	19, 22	bitr4i 277	. . . . . . 7 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))
24	23	imbi2i 336	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
25	2, 3, 24	3bitri 297	. . . . 5 ⊢ (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
26	25	2albii 1823	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
27		ssrel 5693	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
28	26, 27	bitr4id 290	. . 3 ⊢ (Rel 𝐹 → (∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
29	28	pm5.32i 575	. 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30		dffun4 6446	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31		sscoid 34215	. 2 ⊢ (𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
32	29, 30, 31	3bitr4i 303	1 ⊢ (Fun 𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   I cid 5488   ∘ ccom 5593  Rel wrel 5594  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-fun 6435

This theorem is referenced by: (None)