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Theorem gruurn 10066
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10067 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 8269 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈𝑚 𝐴) ↔ 𝐹:𝐴𝑈))
2 elgrug 10060 . . . . . . 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 5688 . . . . . . . . . 10 (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
65unieqd 4755 . . . . . . . . 9 (𝑦 = 𝐹 ran 𝑦 = ran 𝐹)
76eleq1d 2867 . . . . . . . 8 (𝑦 = 𝐹 → ( ran 𝑦𝑈 ran 𝐹𝑈))
87rspccv 3556 . . . . . . 7 (∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈 → (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
983ad2ant3 1128 . . . . . 6 ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈) → (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
109ralimi 3127 . . . . 5 (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
11 oveq2 7024 . . . . . . . 8 (𝑥 = 𝐴 → (𝑈𝑚 𝑥) = (𝑈𝑚 𝐴))
1211eleq2d 2868 . . . . . . 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 3556 . . . . 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 1110 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  𝒫 cpw 4453  {cpr 4474   cuni 4745  Tr wtr 5063  ran crn 5444  wf 6221  (class class class)co 7016  𝑚 cmap 8256  Univcgru 10058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-gru 10059
This theorem is referenced by:  gruiun  10067  grurn  10069  intgru  10082
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