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Theorem gruurn 9564
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9565 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 7815 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈𝑚 𝐴) ↔ 𝐹:𝐴𝑈))
2 elgrug 9558 . . . . . . 7 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))))
32ibi 256 . . . . . 6 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈)))
43simprd 479 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))
5 rneq 5311 . . . . . . . . . 10 (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
65unieqd 4412 . . . . . . . . 9 (𝑦 = 𝐹 ran 𝑦 = ran 𝐹)
76eleq1d 2683 . . . . . . . 8 (𝑦 = 𝐹 → ( ran 𝑦𝑈 ran 𝐹𝑈))
87rspccv 3292 . . . . . . 7 (∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈 → (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
983ad2ant3 1082 . . . . . 6 ((𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈) → (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
109ralimi 2947 . . . . 5 (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
11 oveq2 6612 . . . . . . . 8 (𝑥 = 𝐴 → (𝑈𝑚 𝑥) = (𝑈𝑚 𝐴))
1211eleq2d 2684 . . . . . . 7 (𝑥 = 𝐴 → (𝐹 ∈ (𝑈𝑚 𝑥) ↔ 𝐹 ∈ (𝑈𝑚 𝐴)))
1312imbi1d 331 . . . . . 6 (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈) ↔ (𝐹 ∈ (𝑈𝑚 𝐴) → ran 𝐹𝑈)))
1413rspccv 3292 . . . . 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 445 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈𝑚 𝐴) → ran 𝐹𝑈))
171, 16sylbird 250 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹:𝐴𝑈 ran 𝐹𝑈))
18173impia 1258 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐹:𝐴𝑈) → ran 𝐹𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  𝒫 cpw 4130  {cpr 4150   cuni 4402  Tr wtr 4712  ran crn 5075  wf 5843  (class class class)co 6604  𝑚 cmap 7802  Univcgru 9556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-gru 9557
This theorem is referenced by:  gruiun  9565  grurn  9567  intgru  9580
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