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Theorem funpartfun 35945
Description: The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
funpartfun Fun Funpart𝐹

Proof of Theorem funpartfun
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 6022 . 2 Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2 vex 3483 . . . . . . 7 𝑧 ∈ V
32brresi 6005 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
43simprbi 496 . . . . 5 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧𝑥𝐹𝑧)
5 vex 3483 . . . . . . . 8 𝑦 ∈ V
65brresi 6005 . . . . . . 7 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7 funpartlem 35944 . . . . . . . 8 (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑤(𝐹 “ {𝑥}) = {𝑤})
87anbi1i 624 . . . . . . 7 ((𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦) ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
96, 8bitri 275 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
10 df-br 5143 . . . . . . . . . . 11 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11 df-br 5143 . . . . . . . . . . 11 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1210, 11anbi12i 628 . . . . . . . . . 10 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13 vex 3483 . . . . . . . . . . . 12 𝑥 ∈ V
1413, 5elimasn 6107 . . . . . . . . . . 11 (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1513, 2elimasn 6107 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1614, 15anbi12i 628 . . . . . . . . . 10 ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
1712, 16bitr4i 278 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})))
18 eleq2 2829 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
19 eleq2 2829 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 𝑧 ∈ {𝑤}))
2018, 19anbi12d 632 . . . . . . . . . 10 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤})))
21 velsn 4641 . . . . . . . . . . 11 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22 velsn 4641 . . . . . . . . . . 11 (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23 equtr2 2025 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑧 = 𝑤) → 𝑦 = 𝑧)
2421, 22, 23syl2anb 598 . . . . . . . . . 10 ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
2520, 24biimtrdi 253 . . . . . . . . 9 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
2617, 25biimtrid 242 . . . . . . . 8 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2726exlimiv 1929 . . . . . . 7 (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2827impl 455 . . . . . 6 (((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
299, 28sylanb 581 . . . . 5 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)
304, 29sylan2 593 . . . 4 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3130gen2 1795 . . 3 𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3231ax-gen 1794 . 2 𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
33 df-funpart 35876 . . . 4 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
3433funeqi 6586 . . 3 (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35 dffun2 6570 . . 3 (Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)))
3634, 35bitri 275 . 2 (Fun Funpart𝐹 ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)))
371, 32, 36mpbir2an 711 1 Fun Funpart𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  cin 3949  {csn 4625  cop 4631   class class class wbr 5142   × cxp 5682  dom cdm 5684  cres 5686  cima 5687  ccom 5688  Rel wrel 5689  Fun wfun 6554  Singletoncsingle 35840   Singletons csingles 35841  Imagecimage 35842  Funpartcfunpart 35851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-symdif 4252  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-eprel 5583  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-txp 35856  df-singleton 35864  df-singles 35865  df-image 35866  df-funpart 35876
This theorem is referenced by:  fullfunfnv  35948  fullfunfv  35949
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