Theorem funpartfun 34915

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.

Step	Hyp	Ref	Expression
1		relres 6011	. 2 ⊢ Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		vex 3479	. . . . . . 7 ⊢ 𝑧 ∈ V
3	2	brresi 5991	. . . . . 6 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
4	3	simprbi 498	. . . . 5 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 → 𝑥𝐹𝑧)
5		vex 3479	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	5	brresi 5991	. . . . . . 7 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7		funpartlem 34914	. . . . . . . 8 ⊢ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑤(𝐹 “ {𝑥}) = {𝑤})
8	7	anbi1i 625	. . . . . . 7 ⊢ ((𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦) ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
9	6, 8	bitri 275	. . . . . 6 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
10		df-br 5150	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11		df-br 5150	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
12	10, 11	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
14	13, 5	elimasn 6089	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
15	13, 2	elimasn 6089	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
16	14, 15	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
17	12, 16	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (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}))
20	18, 19	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤})))
21		velsn 4645	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22		velsn 4645	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23		equtr2 2031	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑤) → 𝑦 = 𝑧)
24	21, 22, 23	syl2anb 599	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
25	20, 24	syl6bi 253	. . . . . . . . 9 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
26	17, 25	biimtrid 241	. . . . . . . 8 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
27	26	exlimiv 1934	. . . . . . 7 ⊢ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
28	27	impl 457	. . . . . 6 ⊢ (((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
29	9, 28	sylanb 582	. . . . 5 ⊢ ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
30	4, 29	sylan2 594	. . . 4 ⊢ ((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
31	30	gen2 1799	. . 3 ⊢ ∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
32	31	ax-gen 1798	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)
33		df-funpart 34846	. . . 4 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
34	33	funeqi 6570	. . 3 ⊢ (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35		dffun2 6554	. . 3 ⊢ (Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)))
36	34, 35	bitri 275	. 2 ⊢ (Fun Funpart𝐹 ↔ (Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ∧ 𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧) → 𝑦 = 𝑧)))
37	1, 32, 36	mpbir2an 710	1 ⊢ Fun Funpart𝐹

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948  {csn 4629  ⟨cop 4635   class class class wbr 5149   × cxp 5675  dom cdm 5677   ↾ cres 5679   “ cima 5680   ∘ ccom 5681  Rel wrel 5682  Fun wfun 6538  Singletoncsingle 34810   Singletons csingles 34811  Imagecimage 34812  Funpartcfunpart 34821

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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-singleton 34834  df-singles 34835  df-image 34836  df-funpart 34846

This theorem is referenced by:  fullfunfnv  34918  fullfunfv  34919