Theorem funpartfun 33408

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 5885	. 2 ⊢ Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		vex 3500	. . . . . . 7 ⊢ 𝑧 ∈ V
3	2	brresi 5865	. . . . . 6 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
4	3	simprbi 499	. . . . 5 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 → 𝑥𝐹𝑧)
5		vex 3500	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	5	brresi 5865	. . . . . . 7 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7		funpartlem 33407	. . . . . . . 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 277	. . . . . 6 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦))
10		df-br 5070	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11		df-br 5070	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
12	10, 11	anbi12i 628	. . . . . . . . . 10 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13		vex 3500	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
14	13, 5	elimasn 5957	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
15	13, 2	elimasn 5957	. . . . . . . . . . 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 280	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})))
18		eleq2 2904	. . . . . . . . . . 11 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
19		eleq2 2904	. . . . . . . . . . 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 4586	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22		velsn 4586	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23		equtr2 2033	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑤) → 𝑦 = 𝑧)
24	21, 22, 23	syl2anb 599	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
25	20, 24	syl6bi 255	. . . . . . . . 9 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
26	17, 25	syl5bi 244	. . . . . . . 8 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
27	26	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
28	27	impl 458	. . . . . 6 ⊢ (((∃𝑤(𝐹 “ {𝑥}) = {𝑤} ∧ 𝑥𝐹𝑦) ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)
29	9, 28	sylanb 583	. . . . 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 1796	. . 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 1795	. 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 33339	. . . 4 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
34	33	funeqi 6379	. . 3 ⊢ (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35		dffun2 6368	. . 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 277	. 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 709	1 ⊢ Fun Funpart𝐹

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534   = wceq 1536  ∃wex 1779   ∈ wcel 2113  Vcvv 3497   ∩ cin 3938  {csn 4570  ⟨cop 4576   class class class wbr 5069   × cxp 5556  dom cdm 5558   ↾ cres 5560   “ cima 5561   ∘ ccom 5562  Rel wrel 5563  Fun wfun 6352  Singletoncsingle 33303   Singletons csingles 33304  Imagecimage 33305  Funpartcfunpart 33314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-eprel 5468  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-1st 7692  df-2nd 7693  df-txp 33319  df-singleton 33327  df-singles 33328  df-image 33329  df-funpart 33339

This theorem is referenced by:  fullfunfnv  33411  fullfunfv  33412