Theorem gruurn 10711

Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10712 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

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

Proof of Theorem gruurn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapg 8778	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝑈))
2		elgrug 10705	. . . . . . 7 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
3	2	ibi 267	. . . . . 6 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))
4	3	simprd 495	. . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
5		rneq 5884	. . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
6	5	unieqd 4875	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ∪ ran 𝑦 = ∪ ran 𝐹)
7	6	eleq1d 2820	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∪ ran 𝑦 ∈ 𝑈 ↔ ∪ ran 𝐹 ∈ 𝑈))
8	7	rspccv 3572	. . . . . . 7 ⊢ (∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈))
9	8	3ad2ant3 1136	. . . . . 6 ⊢ ((𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈))
10	9	ralimi 3072	. . . . 5 ⊢ (∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 (𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈))
11		oveq2 7366	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑈 ↑m 𝑥) = (𝑈 ↑m 𝐴))
12	11	eleq2d 2821	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹 ∈ (𝑈 ↑m 𝑥) ↔ 𝐹 ∈ (𝑈 ↑m 𝐴)))
13	12	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈) ↔ (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈)))
14	13	rspccv 3572	. . . . 5 ⊢ (∀𝑥 ∈ 𝑈 (𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈)))
15	4, 10, 14	3syl 18	. . . 4 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈)))
16	15	imp 406	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈))
17	1, 16	sylbird 260	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹:𝐴⟶𝑈 → ∪ ran 𝐹 ∈ 𝑈))
18	17	3impia 1118	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3050  𝒫 cpw 4553  {cpr 4581  ∪ cuni 4862  Tr wtr 5204  ran crn 5624  ⟶wf 6487  (class class class)co 7358   ↑_m cmap 8765  Univcgru 10703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767  df-gru 10704

This theorem is referenced by:  gruiun  10712  grurn  10714  intgru  10727