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Theorem gruurn 10412
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10413 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruurn ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)

Proof of Theorem gruurn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapg 8521 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈m 𝐴) ↔ 𝐹:𝐴𝑈))
2 elgrug 10406 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
32ibi 270 . . . . . 6 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)))
43simprd 499 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
5 rneq 5805 . . . . . . . . . 10 (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
65unieqd 4833 . . . . . . . . 9 (𝑦 = 𝐹 ran 𝑦 = ran 𝐹)
76eleq1d 2822 . . . . . . . 8 (𝑦 = 𝐹 → ( ran 𝑦𝑈 ran 𝐹𝑈))
87rspccv 3534 . . . . . . 7 (∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈 → (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
983ad2ant3 1137 . . . . . 6 ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
109ralimi 3083 . . . . 5 (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
11 oveq2 7221 . . . . . . . 8 (𝑥 = 𝐴 → (𝑈m 𝑥) = (𝑈m 𝐴))
1211eleq2d 2823 . . . . . . 7 (𝑥 = 𝐴 → (𝐹 ∈ (𝑈m 𝑥) ↔ 𝐹 ∈ (𝑈m 𝐴)))
1312imbi1d 345 . . . . . 6 (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈) ↔ (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈)))
1413rspccv 3534 . . . . 5 (∀𝑥𝑈 (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈) → (𝐴𝑈 → (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈)))
154, 10, 143syl 18 . . . 4 (𝑈 ∈ Univ → (𝐴𝑈 → (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈)))
1615imp 410 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈))
171, 16sylbird 263 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹:𝐴𝑈 ran 𝐹𝑈))
18173impia 1119 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  𝒫 cpw 4513  {cpr 4543   cuni 4819  Tr wtr 5161  ran crn 5552  wf 6376  (class class class)co 7213  m cmap 8508  Univcgru 10404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-gru 10405
This theorem is referenced by:  gruiun  10413  grurn  10415  intgru  10428
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