Theorem gruima 10871

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 1192	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → Fun 𝐹)
2		funrel 6595	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
3		df-ima 5713	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
4		resres 6022	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
5		resdm 6055	. . . . . . . 8 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
6	5	reseq1d 6008	. . . . . . 7 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
7	4, 6	eqtr3id 2794	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
8	7	rneqd 5963	. . . . 5 ⊢ (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = ran (𝐹 ↾ 𝐴))
9	3, 8	eqtr4id 2799	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)))
10	1, 2, 9	3syl 18	. . 3 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) = ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)))
11		simpl1 1191	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ Univ)
12		simpr 484	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈)
13		inss2 4259	. . . . . 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 10865	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
16	11, 12, 14, 15	syl3anc 1371	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
17		funforn 6841	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
18		fof 6834	. . . . . . . 8 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
19	17, 18	sylbi 217	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
20		inss1 4258	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
21		fssres 6787	. . . . . . 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 585	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹)
23		ffn 6747	. . . . . 6 ⊢ ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴))
24	1, 22, 23	3syl 18	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) Fn (dom 𝐹 ∩ 𝐴))
25		simpl3 1193	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ⊆ 𝑈)
26	10, 25	eqsstrrd 4048	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈)
27		df-f 6577	. . . . 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 582	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈)
29		grurn 10870	. . . 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 1371	. . 3 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈)
31	10, 30	eqeltrd 2844	. 2 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ∈ 𝑈)
32	31	ex 412	1 ⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Rel wrel 5705  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  Univcgru 10859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-gru 10860

This theorem is referenced by: (None)