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Theorem funpartfun 34574
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 5967 . 2 Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2 vex 3448 . . . . . . 7 𝑧 ∈ V
32brresi 5947 . . . . . 6 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
43simprbi 498 . . . . 5 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧𝑥𝐹𝑧)
5 vex 3448 . . . . . . . 8 𝑦 ∈ V
65brresi 5947 . . . . . . 7 (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7 funpartlem 34573 . . . . . . . 8 (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑤(𝐹 “ {𝑥}) = {𝑤})
87anbi1i 625 . . . . . . 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 5107 . . . . . . . . . . 11 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11 df-br 5107 . . . . . . . . . . 11 (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
1210, 11anbi12i 628 . . . . . . . . . 10 ((𝑥𝐹𝑦𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13 vex 3448 . . . . . . . . . . . 12 𝑥 ∈ V
1413, 5elimasn 6042 . . . . . . . . . . 11 (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
1513, 2elimasn 6042 . . . . . . . . . . 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 2823 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
19 eleq2 2823 . . . . . . . . . . 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 4603 . . . . . . . . . . 11 (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22 velsn 4603 . . . . . . . . . . 11 (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23 equtr2 2031 . . . . . . . . . . 11 ((𝑦 = 𝑤𝑧 = 𝑤) → 𝑦 = 𝑧)
2421, 22, 23syl2anb 599 . . . . . . . . . 10 ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
2520, 24syl6bi 253 . . . . . . . . 9 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
2617, 25biimtrid 241 . . . . . . . 8 ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2726exlimiv 1934 . . . . . . 7 (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧))
2827impl 457 . . . . . 6 (((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
299, 28sylanb 582 . . . . 5 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)
304, 29sylan2 594 . . . 4 ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3130gen2 1799 . . 3 𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
3231ax-gen 1798 . 2 𝑥𝑦𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
33 df-funpart 34505 . . . 4 Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
3433funeqi 6523 . . 3 (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35 dffun2 6507 . . 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 710 1 Fun Funpart𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  cin 3910  {csn 4587  cop 4593   class class class wbr 5106   × cxp 5632  dom cdm 5634  cres 5636  cima 5637  ccom 5638  Rel wrel 5639  Fun wfun 6491  Singletoncsingle 34469   Singletons csingles 34470  Imagecimage 34471  Funpartcfunpart 34480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-symdif 4203  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-eprel 5538  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923  df-txp 34485  df-singleton 34493  df-singles 34494  df-image 34495  df-funpart 34505
This theorem is referenced by:  fullfunfnv  34577  fullfunfv  34578
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