Theorem grurn 10761

Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10759 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 1136	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝑈 ∈ Univ)
2		gruurn 10758	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)
3		grupw 10755	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ ran 𝐹 ∈ 𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
4	1, 2, 3	syl2anc 584	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈)
5		pwuni 4912	. . 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 10756	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 ∪ ran 𝐹 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) → ran 𝐹 ∈ 𝑈)
8	1, 4, 6, 7	syl3anc 1373	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ran crn 5642  ⟶wf 6510  Univcgru 10750

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-gru 10751

This theorem is referenced by:  gruima  10762  gruf  10771  gruen  10772