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Theorem funpartfun 34984
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 6010 . 2 Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2 vex 3478 . . . . . . 7 𝑧 ∈ V
32brresi 5990 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
43simprbi 497 . . . . 5 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧𝑥𝐹𝑧)
5 vex 3478 . . . . . . . 8 𝑦 ∈ V
65brresi 5990 . . . . . . 7 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7 funpartlem 34983 . . . . . . . 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 274 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
10 df-br 5149 . . . . . . . . . . 11 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11 df-br 5149 . . . . . . . . . . 11 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1210, 11anbi12i 627 . . . . . . . . . 10 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13 vex 3478 . . . . . . . . . . . 12 𝑥 ∈ V
1413, 5elimasn 6088 . . . . . . . . . . 11 (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1513, 2elimasn 6088 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1614, 15anbi12i 627 . . . . . . . . . 10 ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
1712, 16bitr4i 277 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})))
18 eleq2 2822 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
19 eleq2 2822 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 𝑧 ∈ {𝑤}))
2018, 19anbi12d 631 . . . . . . . . . 10 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤})))
21 velsn 4644 . . . . . . . . . . 11 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22 velsn 4644 . . . . . . . . . . 11 (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23 equtr2 2030 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑧 = 𝑤) → 𝑦 = 𝑧)
2421, 22, 23syl2anb 598 . . . . . . . . . 10 ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
2520, 24syl6bi 252 . . . . . . . . 9 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
2617, 25biimtrid 241 . . . . . . . 8 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2726exlimiv 1933 . . . . . . 7 (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2827impl 456 . . . . . 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 1798 . . 3 𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3231ax-gen 1797 . 2 𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
33 df-funpart 34915 . . . 4 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
3433funeqi 6569 . . 3 (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35 dffun2 6553 . . 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 274 . 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 709 1 Fun Funpart𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  cin 3947  {csn 4628  cop 4634   class class class wbr 5148   × cxp 5674  dom cdm 5676  cres 5678  cima 5679  ccom 5680  Rel wrel 5681  Fun wfun 6537  Singletoncsingle 34879   Singletons csingles 34880  Imagecimage 34881  Funpartcfunpart 34890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977  df-2nd 7978  df-txp 34895  df-singleton 34903  df-singles 34904  df-image 34905  df-funpart 34915
This theorem is referenced by:  fullfunfnv  34987  fullfunfv  34988
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