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Theorem dffun10 33854
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 impexp 454 . . . . . . 7 (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
21albii 1826 . . . . . 6 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
3 19.21v 1946 . . . . . 6 (∀𝑧(⟨𝑥, 𝑦⟩ ∈ 𝐹 → (⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)))
4 vex 3402 . . . . . . . . . . 11 𝑥 ∈ V
5 vex 3402 . . . . . . . . . . 11 𝑦 ∈ V
64, 5opelco 5714 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦))
7 df-br 5031 . . . . . . . . . . . 12 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
8 brv 5330 . . . . . . . . . . . . . 14 𝑧V𝑦
9 brdif 5083 . . . . . . . . . . . . . 14 (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦))
108, 9mpbiran 709 . . . . . . . . . . . . 13 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦)
115ideq 5695 . . . . . . . . . . . . . 14 (𝑧 I 𝑦𝑧 = 𝑦)
12 equcom 2030 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝑦 = 𝑧)
1311, 12bitri 278 . . . . . . . . . . . . 13 (𝑧 I 𝑦𝑦 = 𝑧)
1410, 13xchbinx 337 . . . . . . . . . . . 12 (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)
157, 14anbi12i 630 . . . . . . . . . . 11 ((𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
1615exbii 1854 . . . . . . . . . 10 (∃𝑧(𝑥𝐹𝑧𝑧(V ∖ I )𝑦) ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧))
17 exanali 1866 . . . . . . . . . 10 (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹 ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
186, 16, 173bitri 300 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹) ↔ ¬ ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧))
1918con2bii 361 . . . . . . . 8 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
20 opex 5322 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
21 eldif 3853 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹)))
2220, 21mpbiran 709 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ¬ ⟨𝑥, 𝑦⟩ ∈ ((V ∖ I ) ∘ 𝐹))
2319, 22bitr4i 281 . . . . . . 7 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧) ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))
2423imbi2i 339 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∀𝑧(⟨𝑥, 𝑧⟩ ∈ 𝐹𝑦 = 𝑧)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
252, 3, 243bitri 300 . . . . 5 (∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
26252albii 1827 . . . 4 (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹))))
27 ssrel 5628 . . . 4 (Rel 𝐹 → (𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (V ∖ ((V ∖ I ) ∘ 𝐹)))))
2826, 27bitr4id 293 . . 3 (Rel 𝐹 → (∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧) ↔ 𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
2928pm5.32i 578 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
30 dffun4 6351 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
31 sscoid 33853 . 2 (𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))) ↔ (Rel 𝐹𝐹 ⊆ (V ∖ ((V ∖ I ) ∘ 𝐹))))
3229, 30, 313bitr4i 306 1 (Fun 𝐹𝐹 ⊆ ( I ∘ (V ∖ ((V ∖ I ) ∘ 𝐹))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1540  wex 1786  wcel 2114  Vcvv 3398  cdif 3840  wss 3843  cop 4522   class class class wbr 5030   I cid 5428  ccom 5529  Rel wrel 5530  Fun wfun 6333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-fun 6341
This theorem is referenced by: (None)
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