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Theorem gruima 9503
 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 1058 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → Fun 𝐹)
2 funrel 5821 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 resres 5329 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
4 resdm 5361 . . . . . . . 8 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
54reseq1d 5316 . . . . . . 7 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
63, 5syl5eqr 2658 . . . . . 6 (Rel 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
76rneqd 5274 . . . . 5 (Rel 𝐹 → ran (𝐹 ↾ (dom 𝐹𝐴)) = ran (𝐹𝐴))
8 df-ima 5051 . . . . 5 (𝐹𝐴) = ran (𝐹𝐴)
97, 8syl6reqr 2663 . . . 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 1057 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝑈 ∈ Univ)
12 simpr 476 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → 𝐴𝑈)
13 inss2 3796 . . . . . 6 (dom 𝐹𝐴) ⊆ 𝐴
1413a1i 11 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ⊆ 𝐴)
15 gruss 9497 . . . . 5 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ (dom 𝐹𝐴) ⊆ 𝐴) → (dom 𝐹𝐴) ∈ 𝑈)
1611, 12, 14, 15syl3anc 1318 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (dom 𝐹𝐴) ∈ 𝑈)
17 funforn 6035 . . . . . . . 8 (Fun 𝐹𝐹:dom 𝐹onto→ran 𝐹)
18 fof 6028 . . . . . . . 8 (𝐹:dom 𝐹onto→ran 𝐹𝐹:dom 𝐹⟶ran 𝐹)
1917, 18sylbi 206 . . . . . . 7 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
20 inss1 3795 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
21 fssres 5983 . . . . . . 7 ((𝐹:dom 𝐹⟶ran 𝐹 ∧ (dom 𝐹𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
2219, 20, 21sylancl 693 . . . . . 6 (Fun 𝐹 → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ran 𝐹)
23 ffn 5958 . . . . . 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 1059 . . . . . 6 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹𝐴) ⊆ 𝑈)
2610, 25eqsstr3d 3603 . . . . 5 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈)
27 df-f 5808 . . . . 5 ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈 ↔ ((𝐹 ↾ (dom 𝐹𝐴)) Fn (dom 𝐹𝐴) ∧ ran (𝐹 ↾ (dom 𝐹𝐴)) ⊆ 𝑈))
2824, 26, 27sylanbrc 695 . . . 4 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈)
29 grurn 9502 . . . 4 ((𝑈 ∈ Univ ∧ (dom 𝐹𝐴) ∈ 𝑈 ∧ (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3011, 16, 28, 29syl3anc 1318 . . 3 (((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹𝐴) ⊆ 𝑈) ∧ 𝐴𝑈) → ran (𝐹 ↾ (dom 𝐹𝐴)) ∈ 𝑈)
3110, 30eqeltrd 2688 . 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 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  Univcgru 9491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-gru 9492 This theorem is referenced by: (None)
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