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Theorem dffun10 32358
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 5347 . . . 4 (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2 impexp 437 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
32albii 1895 . . . . . 6 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
4 19.21v 2020 . . . . . 6 (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
5 vex 3354 . . . . . . . . . . 11 𝑥 ∈ V
6 vex 3354 . . . . . . . . . . 11 𝑦 ∈ V
75, 6opelco 5432 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦))
8 df-br 4787 . . . . . . . . . . . 12 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
9 brv 5068 . . . . . . . . . . . . . 14 𝑧V𝑦
10 brdif 4839 . . . . . . . . . . . . . 14 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
119, 10mpbiran 688 . . . . . . . . . . . . 13 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
126ideq 5413 . . . . . . . . . . . . . 14 (𝑧 I 𝑦𝑧 = 𝑦)
13 equcom 2103 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝑦 = 𝑧)
1412, 13bitri 264 . . . . . . . . . . . . 13 (𝑧 I 𝑦𝑦 = 𝑧)
1511, 14xchbinx 323 . . . . . . . . . . . 12 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
168, 15anbi12i 612 . . . . . . . . . . 11 ((𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
1716exbii 1924 . . . . . . . . . 10 (∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
18 exanali 1937 . . . . . . . . . 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 5060 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
22 eldif 3733 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
2321, 22mpbiran 688 . . . . . . . 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 1896 . . . 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 564 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30 dffun4 6043 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31 sscoid 32357 . 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 382  wal 1629  wex 1852  wcel 2145  Vcvv 3351  cdif 3720  wss 3723  cop 4322   class class class wbr 4786   I cid 5156  ccom 5253  Rel wrel 5254  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-fun 6033
This theorem is referenced by: (None)
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