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Theorem gruurn 10208
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10209 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 8408 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈m 𝐴) ↔ 𝐹:𝐴𝑈))
2 elgrug 10202 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
32ibi 268 . . . . . 6 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)))
43simprd 496 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
5 rneq 5799 . . . . . . . . . 10 (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
65unieqd 4840 . . . . . . . . 9 (𝑦 = 𝐹 ran 𝑦 = ran 𝐹)
76eleq1d 2894 . . . . . . . 8 (𝑦 = 𝐹 → ( ran 𝑦𝑈 ran 𝐹𝑈))
87rspccv 3617 . . . . . . 7 (∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈 → (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
983ad2ant3 1127 . . . . . 6 ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
109ralimi 3157 . . . . 5 (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 (𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈))
11 oveq2 7153 . . . . . . . 8 (𝑥 = 𝐴 → (𝑈m 𝑥) = (𝑈m 𝐴))
1211eleq2d 2895 . . . . . . 7 (𝑥 = 𝐴 → (𝐹 ∈ (𝑈m 𝑥) ↔ 𝐹 ∈ (𝑈m 𝐴)))
1312imbi1d 343 . . . . . 6 (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈m 𝑥) → ran 𝐹𝑈) ↔ (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈)))
1413rspccv 3617 . . . . 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 407 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈m 𝐴) → ran 𝐹𝑈))
171, 16sylbird 261 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹:𝐴𝑈 ran 𝐹𝑈))
18173impia 1109 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  𝒫 cpw 4535  {cpr 4559   cuni 4830  Tr wtr 5163  ran crn 5549  wf 6344  (class class class)co 7145  m cmap 8395  Univcgru 10200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-gru 10201
This theorem is referenced by:  gruiun  10209  grurn  10211  intgru  10224
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