Theorem gruima 10221

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 1187	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → Fun 𝐹)
2		funrel 6369	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
3		resres 5863	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
4		resdm 5894	. . . . . . . 8 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
5	4	reseq1d 5849	. . . . . . 7 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
6	3, 5	syl5eqr 2869	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
7	6	rneqd 5805	. . . . 5 ⊢ (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = ran (𝐹 ↾ 𝐴))
8		df-ima 5565	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
9	7, 8	syl6reqr 2874	. . . 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 1186	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ Univ)
12		simpr 487	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈)
13		inss2 4203	. . . . . 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 10215	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (dom 𝐹 ∩ 𝐴) ⊆ 𝐴) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
16	11, 12, 14, 15	syl3anc 1366	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (dom 𝐹 ∩ 𝐴) ∈ 𝑈)
17		funforn 6594	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
18		fof 6587	. . . . . . . 8 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
19	17, 18	sylbi 219	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
20		inss1 4202	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
21		fssres 6541	. . . . . . 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 588	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ran 𝐹)
23		ffn 6511	. . . . . 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 1188	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ⊆ 𝑈)
26	10, 25	eqsstrrd 4003	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ⊆ 𝑈)
27		df-f 6356	. . . . 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 585	. . . 4 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶𝑈)
29		grurn 10220	. . . 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 1366	. . 3 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → ran (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) ∈ 𝑈)
31	10, 30	eqeltrd 2912	. 2 ⊢ (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) ∧ 𝐴 ∈ 𝑈) → (𝐹 “ 𝐴) ∈ 𝑈)
32	31	ex 415	1 ⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   ∩ cin 3932   ⊆ wss 3933  dom cdm 5552  ran crn 5553   ↾ cres 5554   “ cima 5555  Rel wrel 5557  Fun wfun 6346   Fn wfn 6347  ⟶wf 6348  –onto→wfo 6350  Univcgru 10209

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 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-tr 5170  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-ov 7156  df-oprab 7157  df-mpo 7158  df-map 8405  df-gru 10210

This theorem is referenced by: (None)