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Theorem gruima 10779
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 1192 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → Fun 𝐹)
2 funrel 6554 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 df-ima 5682 . . . . 5 (𝐹𝐴) = ran (𝐹𝐴)
4 resres 5986 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
5 resdm 6018 . . . . . . . 8 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
65reseq1d 5972 . . . . . . 7 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
74, 6eqtr3id 2785 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
87rneqd 5929 . . . . 5 (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹𝐴)) = ran (𝐹𝐴))
93, 8eqtr4id 2790 . . . 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 1191 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝑈 ∈ Univ)
12 simpr 485 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝐴𝑈)
13 inss2 4225 . . . . . 6 (dom 𝐹𝐴) ⊆ 𝐴
1413a1i 11 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ⊆ 𝐴)
15 gruss 10773 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (dom 𝐹𝐴) ⊆ 𝐴) → (dom 𝐹𝐴) ∈ 𝑈)
1611, 12, 14, 15syl3anc 1371 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ∈ 𝑈)
17 funforn 6799 . . . . . . . 8 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
18 fof 6792 . . . . . . . 8 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
1917, 18sylbi 216 . . . . . . 7 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
20 inss1 4224 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
21 fssres 6744 . . . . . . 7 ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
2219, 20, 21sylancl 586 . . . . . 6 (Fun 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
23 ffn 6704 . . . . . 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 1193 . . . . . 6 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ⊆ 𝑈)
2610, 25eqsstrrd 4017 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈)
27 df-f 6536 . . . . 5 ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹𝐴)) Fn (dom 𝐹𝐴) ∧ ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈))
2824, 26, 27sylanbrc 583 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈)
29 grurn 10778 . . . 4 ((𝑈 ∈ Univ ∧ (dom 𝐹𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3011, 16, 28, 29syl3anc 1371 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3110, 30eqeltrd 2832 . 2 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ∈ 𝑈)
3231ex 413 1 ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3943  wss 3944  dom cdm 5669  ran crn 5670  cres 5671  cima 5672  Rel wrel 5674  Fun wfun 6526   Fn wfn 6527  wf 6528  ontowfo 6530  Univcgru 10767
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-gru 10768
This theorem is referenced by: (None)
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