MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruima Structured version   Visualization version   GIF version

Theorem gruima 9662
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 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))

Proof of Theorem gruima
StepHypRef Expression
1 simpl2 1085 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → Fun 𝐹)
2 funrel 5943 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 resres 5444 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
4 resdm 5476 . . . . . . . 8 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
54reseq1d 5427 . . . . . . 7 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
63, 5syl5eqr 2699 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
76rneqd 5385 . . . . 5 (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹𝐴)) = ran (𝐹𝐴))
8 df-ima 5156 . . . . 5 (𝐹𝐴) = ran (𝐹𝐴)
97, 8syl6reqr 2704 . . . 4 (Rel 𝐹 → (𝐹𝐴) = ran (𝐹 ↾ (dom 𝐹𝐴)))
101, 2, 93syl 18 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) = ran (𝐹 ↾ (dom 𝐹𝐴)))
11 simpl1 1084 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝑈 ∈ Univ)
12 simpr 476 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝐴𝑈)
13 inss2 3867 . . . . . 6 (dom 𝐹𝐴) ⊆ 𝐴
1413a1i 11 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ⊆ 𝐴)
15 gruss 9656 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (dom 𝐹𝐴) ⊆ 𝐴) → (dom 𝐹𝐴) ∈ 𝑈)
1611, 12, 14, 15syl3anc 1366 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ∈ 𝑈)
17 funforn 6160 . . . . . . . 8 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
18 fof 6153 . . . . . . . 8 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
1917, 18sylbi 207 . . . . . . 7 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
20 inss1 3866 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
21 fssres 6108 . . . . . . 7 ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
2219, 20, 21sylancl 695 . . . . . 6 (Fun 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
23 ffn 6083 . . . . . 6 ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) Fn (dom 𝐹𝐴))
241, 22, 233syl 18 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹 ↾ (dom 𝐹𝐴)) Fn (dom 𝐹𝐴))
25 simpl3 1086 . . . . . 6 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ⊆ 𝑈)
2610, 25eqsstr3d 3673 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈)
27 df-f 5930 . . . . 5 ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹𝐴)) Fn (dom 𝐹𝐴) ∧ ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈))
2824, 26, 27sylanbrc 699 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈)
29 grurn 9661 . . . 4 ((𝑈 ∈ Univ ∧ (dom 𝐹𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3011, 16, 28, 29syl3anc 1366 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3110, 30eqeltrd 2730 . 2 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ∈ 𝑈)
3231ex 449 1 ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  wss 3607  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Rel wrel 5148  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  Univcgru 9650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-gru 9651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator