Theorem funpartfun 36141

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 5964	. 2 ⊢ Rel (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
2		vex 3434	. . . . . . 7 ⊢ 𝑧 ∈ V
3	2	brresi 5947	. . . . . 6 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑧))
4	3	simprbi 497	. . . . 5 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑧 → 𝑥𝐹𝑧)
5		vex 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	5	brresi 5947	. . . . . . 7 ⊢ (𝑥(𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))𝑦 ↔ (𝑥 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ∧ 𝑥𝐹𝑦))
7		funpartlem 36140	. . . . . . . 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 5087	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
11		df-br 5087	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
12	10, 11	anbi12i 629	. . . . . . . . . 10 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
13		vex 3434	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
14	13, 5	elimasn 6049	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
15	13, 2	elimasn 6049	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐹 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐹)
16	14, 15	anbi12i 629	. . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → (𝑦 ∈ (𝐹 “ {𝑥}) ↔ 𝑦 ∈ {𝑤}))
19		eleq2 2826	. . . . . . . . . . 11 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → (𝑧 ∈ (𝐹 “ {𝑥}) ↔ 𝑧 ∈ {𝑤}))
20	18, 19	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) ↔ (𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤})))
21		velsn 4584	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
22		velsn 4584	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {𝑤} ↔ 𝑧 = 𝑤)
23		equtr2 2029	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑤) → 𝑦 = 𝑧)
24	21, 22, 23	syl2anb 599	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑤} ∧ 𝑧 ∈ {𝑤}) → 𝑦 = 𝑧)
25	20, 24	biimtrdi 253	. . . . . . . . 9 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑦 ∈ (𝐹 “ {𝑥}) ∧ 𝑧 ∈ (𝐹 “ {𝑥})) → 𝑦 = 𝑧))
26	17, 25	biimtrid 242	. . . . . . . 8 ⊢ ((𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
27	26	exlimiv 1932	. . . . . . 7 ⊢ (∃𝑤(𝐹 “ {𝑥}) = {𝑤} → ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧))
28	27	impl 455	. . . . . 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 1798	. . 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 1797	. 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 36070	. . . 4 ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
34	33	funeqi 6513	. . 3 ⊢ (Fun Funpart𝐹 ↔ Fun (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
35		dffun2 6502	. . 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 712	1 ⊢ Fun Funpart𝐹

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  {csn 4568  ⟨cop 4574   class class class wbr 5086   × cxp 5622  dom cdm 5624   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  Rel wrel 5629  Fun wfun 6486  Singletoncsingle 36034   Singletons csingles 36035  Imagecimage 36036  Funpartcfunpart 36045

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7935  df-2nd 7936  df-txp 36050  df-singleton 36058  df-singles 36059  df-image 36060  df-funpart 36070

This theorem is referenced by:  fullfunfnv  36144  fullfunfv  36145