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Theorem dffun10 32319
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 5356 . . . 4 (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2 impexp 461 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
32albii 1888 . . . . . 6 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
4 19.21v 2009 . . . . . 6 (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
5 vex 3335 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3335 . . . . . . . . . . 11 𝑦 ∈ V
75, 6opelco 5441 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦))
8 df-br 4797 . . . . . . . . . . . 12 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
9 brv 5081 . . . . . . . . . . . . . 14 𝑧V𝑦
10 brdif 4849 . . . . . . . . . . . . . 14 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
119, 10mpbiran 991 . . . . . . . . . . . . 13 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
126ideq 5422 . . . . . . . . . . . . . 14 (𝑧 I 𝑦𝑧 = 𝑦)
13 equcom 2092 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝑦 = 𝑧)
1412, 13bitri 264 . . . . . . . . . . . . 13 (𝑧 I 𝑦𝑦 = 𝑧)
1511, 14xchbinx 323 . . . . . . . . . . . 12 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
168, 15anbi12i 735 . . . . . . . . . . 11 ((𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
1716exbii 1915 . . . . . . . . . 10 (∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
18 exanali 1927 . . . . . . . . . 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 346 . . . . . . . 8 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
21 opex 5073 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
22 eldif 3717 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
2321, 22mpbiran 991 . . . . . . . 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 325 . . . . . 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 1889 . . . 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 672 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30 dffun4 6053 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31 sscoid 32318 . 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 383  wal 1622  wex 1845  wcel 2131  Vcvv 3332  cdif 3704  wss 3707  cop 4319   class class class wbr 4796   I cid 5165  ccom 5262  Rel wrel 5263  Fun wfun 6035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-fun 6043
This theorem is referenced by: (None)
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