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

Theorem gruima 10842
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 1193 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → Fun 𝐹)
2 funrel 6583 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 df-ima 5698 . . . . 5 (𝐹𝐴) = ran (𝐹𝐴)
4 resres 6010 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
5 resdm 6044 . . . . . . . 8 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
65reseq1d 5996 . . . . . . 7 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
74, 6eqtr3id 2791 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
87rneqd 5949 . . . . 5 (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹𝐴)) = ran (𝐹𝐴))
93, 8eqtr4id 2796 . . . 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 1192 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝑈 ∈ Univ)
12 simpr 484 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝐴𝑈)
13 inss2 4238 . . . . . 6 (dom 𝐹𝐴) ⊆ 𝐴
1413a1i 11 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ⊆ 𝐴)
15 gruss 10836 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (dom 𝐹𝐴) ⊆ 𝐴) → (dom 𝐹𝐴) ∈ 𝑈)
1611, 12, 14, 15syl3anc 1373 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ∈ 𝑈)
17 funforn 6827 . . . . . . . 8 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
18 fof 6820 . . . . . . . 8 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
1917, 18sylbi 217 . . . . . . 7 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
20 inss1 4237 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
21 fssres 6774 . . . . . . 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 6736 . . . . . 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 1194 . . . . . 6 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ⊆ 𝑈)
2610, 25eqsstrrd 4019 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈)
27 df-f 6565 . . . . 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 10841 . . . 4 ((𝑈 ∈ Univ ∧ (dom 𝐹𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3011, 16, 28, 29syl3anc 1373 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3110, 30eqeltrd 2841 . 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 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  Univcgru 10830
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-gru 10831
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator