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Theorem funpartfun 31719
Description: The functional part of 𝐹 is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
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 5390 . 2 Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2 vex 3192 . . . . . . 7 𝑧 ∈ V
32brres 5367 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥𝐹𝑧𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
43simplbi 476 . . . . 5 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧𝑥𝐹𝑧)
5 vex 3192 . . . . . . . 8 𝑦 ∈ V
65brres 5367 . . . . . . 7 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
7 ancom 466 . . . . . . . 8 ((𝑥𝐹𝑦𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
8 funpartlem 31718 . . . . . . . . 9 (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑤(𝐹 “ {𝑥}) = {𝑤})
98anbi1i 730 . . . . . . . 8 ((𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦) ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
107, 9bitri 264 . . . . . . 7 ((𝑥𝐹𝑦𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
116, 10bitri 264 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
12 df-br 4619 . . . . . . . . . . 11 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
13 df-br 4619 . . . . . . . . . . 11 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1412, 13anbi12i 732 . . . . . . . . . 10 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
15 vex 3192 . . . . . . . . . . . 12 𝑥 ∈ V
1615, 5elimasn 5454 . . . . . . . . . . 11 (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1715, 2elimasn 5454 . . . . . . . . . . 11 (𝑧 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1816, 17anbi12i 732 . . . . . . . . . 10 ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
1914, 18bitr4i 267 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})))
20 eleq2 2687 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
21 eleq2 2687 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 𝑧 ∈ {𝑤}))
2220, 21anbi12d 746 . . . . . . . . . 10 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤})))
23 velsn 4169 . . . . . . . . . . 11 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
24 velsn 4169 . . . . . . . . . . 11 (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
25 equtr2 1951 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑧 = 𝑤) → 𝑦 = 𝑧)
2623, 24, 25syl2anb 496 . . . . . . . . . 10 ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
2722, 26syl6bi 243 . . . . . . . . 9 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
2819, 27syl5bi 232 . . . . . . . 8 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2928exlimiv 1855 . . . . . . 7 (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
3029impl 649 . . . . . 6 (((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
3111, 30sylanb 489 . . . . 5 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)
324, 31sylan2 491 . . . 4 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3332gen2 1720 . . 3 𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3433ax-gen 1719 . 2 𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
35 df-funpart 31649 . . . 4 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
3635funeqi 5873 . . 3 (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
37 dffun2 5862 . . 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)))
3836, 37bitri 264 . 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)))
391, 34, 38mpbir2an 954 1 Fun Funpart𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cin 3558  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  dom cdm 5079  cres 5081  cima 5082  ccom 5083  Rel wrel 5084  Fun wfun 5846  Singletoncsingle 31613   Singletons csingles 31614  Imagecimage 31615  Funpartcfunpart 31624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31629  df-singleton 31637  df-singles 31638  df-image 31639  df-funpart 31649
This theorem is referenced by:  fullfunfnv  31722  fullfunfv  31723
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