Theorem grurn 9960

Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9958 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 1127	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝑈 ∈ Univ)
2		gruurn 9957	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
3		grupw 9954	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ ran 𝐹 ∈ 𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
4	1, 2, 3	syl2anc 579	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
5		pwuni 4711	. . 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 9955	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 ∪ ran 𝐹 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) → ran 𝐹 ∈ 𝑈)
8	1, 4, 6, 7	syl3anc 1439	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1071   ∈ wcel 2107   ⊆ wss 3792  𝒫 cpw 4379  ∪ cuni 4673  ran crn 5358  ⟶wf 6133  Univcgru 9949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-gru 9950

This theorem is referenced by:  gruima  9961  gruf  9970  gruen  9971