Theorem grurn 10797

Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10795 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grurn	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)

Proof of Theorem grurn

Step	Hyp	Ref	Expression
1		simp1 1154	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝑈 ∈ Univ)
2		gruurn 10794	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
3		grupw 10791	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ ran 𝐹 ∈ 𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
4	1, 2, 3	syl2anc 596	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
5		pwuni 4913	. . 3 ⊢ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹
6	5	a1i 11	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹)
7		gruss 10792	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 ∪ ran 𝐹 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) → ran 𝐹 ∈ 𝑈)
8	1, 4, 6, 7	syl3anc 1398	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874  ran crn 5664  ⟶wf 6536  Univcgru 10786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8828  df-gru 10787

This theorem is used by:  gruima  10798  gruf  10807  gruen  10808