Theorem gruima 10802

Description: A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruima	⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))

Proof of Theorem gruima

Step	Hyp	Ref	Expression
1		simpl2 1211	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → Fun 𝐹)
2		funrel 6557	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
3		df-ima 5676	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
4		resres 5993	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
5		resdm 6027	. . . . . . . 8 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
6	5	reseq1d 5979	. . . . . . 7 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
7	4, 6	eqtr3id 2814	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
8	7	rneqd 5930	. . . . 5 ⊢ (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = ran (𝐹 ↾ 𝐴))
9	3, 8	eqtr4id 2819	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)))
10	1, 2, 9	3syl 19	. . 3 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)))
11		simpl1 1210	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ Univ)
12		simpr 490	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈)
13		inss2 4190	. . . . . 6 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴
14	13	a1i 11	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ⊆ 𝐴)
15		gruss 10796	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
16	11, 12, 14, 15	syl3anc 1398	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
17		funforn 6803	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
18		fof 6796	. . . . . . . 8 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
19	17, 18	sylbi 220	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
20		inss1 4189	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
21		fssres 6748	. . . . . . 7 ⊢ ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹)
22	19, 20, 21	sylancl 598	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹)
23		ffn 6709	. . . . . 6 ⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴))
24	1, 22, 23	3syl 19	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴))
25		simpl3 1212	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ⊆ 𝑈)
26	10, 25	eqsstrrd 3973	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈)
27		df-f 6544	. . . . 5 ⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴) ∧ ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈))
28	24, 26, 27	sylanbrc 595	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈)
29		grurn 10801	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ (dom 𝐹 ∩ 𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈)
30	11, 16, 28, 29	syl3anc 1398	. . 3 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈)
31	10, 30	eqeltrd 2865	. 2 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ∈ 𝑈)
32	31	ex 418	1 ⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∩ cin 3905   ⊆ wss 3906  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Rel wrel 5668  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  Univcgru 10790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-gru 10791

This theorem is used by: (None)