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Theorem dffun10 31655
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 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 ssrel 5173 . . . 4 (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2 impexp 462 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
32albii 1744 . . . . . 6 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
4 19.21v 1870 . . . . . 6 (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
5 vex 3194 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3194 . . . . . . . . . . 11 𝑦 ∈ V
75, 6opelco 5258 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦))
8 df-br 4619 . . . . . . . . . . . 12 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
9 brv 31618 . . . . . . . . . . . . . 14 𝑧V𝑦
10 brdif 4670 . . . . . . . . . . . . . 14 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
119, 10mpbiran 952 . . . . . . . . . . . . 13 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
126ideq 5239 . . . . . . . . . . . . . 14 (𝑧 I 𝑦𝑧 = 𝑦)
13 equcom 1947 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝑦 = 𝑧)
1412, 13bitri 264 . . . . . . . . . . . . 13 (𝑧 I 𝑦𝑦 = 𝑧)
1511, 14xchbinx 324 . . . . . . . . . . . 12 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
168, 15anbi12i 732 . . . . . . . . . . 11 ((𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
1716exbii 1772 . . . . . . . . . 10 (∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
18 exanali 1784 . . . . . . . . . 10 (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
197, 17, 183bitri 286 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
2019con2bii 347 . . . . . . . 8 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
21 opex 4898 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
22 eldif 3570 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
2321, 22mpbiran 952 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
2420, 23bitr4i 267 . . . . . . 7 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))
2524imbi2i 326 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
263, 4, 253bitri 286 . . . . 5 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
27262albii 1745 . . . 4 (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
281, 27syl6rbbr 279 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
2928pm5.32i 668 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30 dffun4 5862 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31 sscoid 31654 . 2 (𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
3229, 30, 313bitr4i 292 1 (Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701  wcel 1992  Vcvv 3191  cdif 3557  wss 3560  cop 4159   class class class wbr 4618   I cid 4989  ccom 5083  Rel wrel 5084  Fun wfun 5844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5852
This theorem is referenced by: (None)
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