Theorem dffun10 33379

Description: Another potential definition of functionhood. 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		ssrel 5660	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2		impexp 453	. . . . . . 7 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
3	2	albii 1819	. . . . . 6 ⊢ (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
4		19.21v 1939	. . . . . 6 ⊢ (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)))
5		vex 3500	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
6		vex 3500	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	5, 6	opelco 5745	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦))
8		df-br 5070	. . . . . . . . . . . 12 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
9		brv 5367	. . . . . . . . . . . . . 14 ⊢ 𝑧V𝑦
10		brdif 5122	. . . . . . . . . . . . . 14 ⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
11	9, 10	mpbiran 707	. . . . . . . . . . . . 13 ⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
12	6	ideq 5726	. . . . . . . . . . . . . 14 ⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦)
13		equcom 2024	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
14	12, 13	bitri 277	. . . . . . . . . . . . 13 ⊢ (𝑧 I 𝑦 ↔ 𝑦 = 𝑧)
15	11, 14	xchbinx 336	. . . . . . . . . . . 12 ⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
16	8, 15	anbi12i 628	. . . . . . . . . . 11 ⊢ ((𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
17	16	exbii 1847	. . . . . . . . . 10 ⊢ (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
18		exanali 1858	. . . . . . . . . 10 ⊢ (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))
19	7, 17, 18	3bitri 299	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧))
20	19	con2bii 360	. . . . . . . 8 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
21		opex 5359	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
22		eldif 3949	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
23	21, 22	mpbiran 707	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
24	20, 23	bitr4i 280	. . . . . . 7 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))
25	24	imbi2i 338	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
26	3, 4, 25	3bitri 299	. . . . 5 ⊢ (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
27	26	2albii 1820	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
28	1, 27	syl6rbbr 292	. . 3 ⊢ (Rel 𝐹 → (∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
29	28	pm5.32i 577	. 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30		dffun4 6370	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31		sscoid 33378	. 2 ⊢ (𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹 ∧ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
32	29, 30, 31	3bitr4i 305	1 ⊢ (Fun 𝐹 ↔ 𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779   ∈ wcel 2113  Vcvv 3497   ∖ cdif 3936   ⊆ wss 3939  ⟨cop 4576   class class class wbr 5069   I cid 5462   ∘ ccom 5562  Rel wrel 5563  Fun wfun 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-fun 6360

This theorem is referenced by: (None)