Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffun10 Structured version   Visualization version   GIF version

Theorem dffun10 35190
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 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))

Proof of Theorem dffun10
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 impexp 449 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
21albii 1819 . . . . . 6 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
3 19.21v 1940 . . . . . 6 (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
4 vex 3476 . . . . . . . . . . 11 𝑥 ∈ V
5 vex 3476 . . . . . . . . . . 11 𝑦 ∈ V
64, 5opelco 5870 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦))
7 df-br 5148 . . . . . . . . . . . 12 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
8 brv 5471 . . . . . . . . . . . . . 14 𝑧V𝑦
9 brdif 5200 . . . . . . . . . . . . . 14 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
108, 9mpbiran 705 . . . . . . . . . . . . 13 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
115ideq 5851 . . . . . . . . . . . . . 14 (𝑧 I 𝑦𝑧 = 𝑦)
12 equcom 2019 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝑦 = 𝑧)
1311, 12bitri 274 . . . . . . . . . . . . 13 (𝑧 I 𝑦𝑦 = 𝑧)
1410, 13xchbinx 333 . . . . . . . . . . . 12 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
157, 14anbi12i 625 . . . . . . . . . . 11 ((𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
1615exbii 1848 . . . . . . . . . 10 (∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
17 exanali 1860 . . . . . . . . . 10 (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
186, 16, 173bitri 296 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
1918con2bii 356 . . . . . . . 8 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
20 opex 5463 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
21 eldif 3957 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
2220, 21mpbiran 705 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
2319, 22bitr4i 277 . . . . . . 7 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))
2423imbi2i 335 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
252, 3, 243bitri 296 . . . . 5 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
26252albii 1820 . . . 4 (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
27 ssrel 5781 . . . 4 (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2826, 27bitr4id 289 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
2928pm5.32i 573 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30 dffun4 6558 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31 sscoid 35189 . 2 (𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
3229, 30, 313bitr4i 302 1 (Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1537  wex 1779  wcel 2104  Vcvv 3472  cdif 3944  wss 3947  cop 4633   class class class wbr 5147   I cid 5572  ccom 5679  Rel wrel 5680  Fun wfun 6536
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-fun 6544
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator