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Theorem gruima 10840
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 1191 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → Fun 𝐹)
2 funrel 6585 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 df-ima 5702 . . . . 5 (𝐹𝐴) = ran (𝐹𝐴)
4 resres 6013 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
5 resdm 6046 . . . . . . . 8 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
65reseq1d 5999 . . . . . . 7 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
74, 6eqtr3id 2789 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
87rneqd 5952 . . . . 5 (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹𝐴)) = ran (𝐹𝐴))
93, 8eqtr4id 2794 . . . 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 1190 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝑈 ∈ Univ)
12 simpr 484 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝐴𝑈)
13 inss2 4246 . . . . . 6 (dom 𝐹𝐴) ⊆ 𝐴
1413a1i 11 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ⊆ 𝐴)
15 gruss 10834 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (dom 𝐹𝐴) ⊆ 𝐴) → (dom 𝐹𝐴) ∈ 𝑈)
1611, 12, 14, 15syl3anc 1370 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ∈ 𝑈)
17 funforn 6828 . . . . . . . 8 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
18 fof 6821 . . . . . . . 8 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
1917, 18sylbi 217 . . . . . . 7 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
20 inss1 4245 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
21 fssres 6775 . . . . . . 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 6737 . . . . . 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 1192 . . . . . 6 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ⊆ 𝑈)
2610, 25eqsstrrd 4035 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈)
27 df-f 6567 . . . . 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 10839 . . . 4 ((𝑈 ∈ Univ ∧ (dom 𝐹𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3011, 16, 28, 29syl3anc 1370 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3110, 30eqeltrd 2839 . 2 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ∈ 𝑈)
3231ex 412 1 ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) → (𝐴𝑈 → (𝐹𝐴) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-gru 10829
This theorem is referenced by: (None)
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