MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruurn Structured version   Visualization version   GIF version

Theorem gruurn 9783
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9784 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 8024 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈𝑚 𝐴) ↔ 𝐹:𝐴𝑈))
2 elgrug 9777 . . . . . . 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 482 . . . . 5 (𝑈 ∈ Univ → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))
5 rneq 5494 . . . . . . . . . 10 (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹)
65unieqd 4586 . . . . . . . . 9 (𝑦 = 𝐹 ran 𝑦 = ran 𝐹)
76eleq1d 2812 . . . . . . . 8 (𝑦 = 𝐹 → ( ran 𝑦𝑈 ran 𝐹𝑈))
87rspccv 3434 . . . . . . 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 3078 . . . . 5 (∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈) → ∀𝑥𝑈 (𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈))
11 oveq2 6809 . . . . . . . 8 (𝑥 = 𝐴 → (𝑈𝑚 𝑥) = (𝑈𝑚 𝐴))
1211eleq2d 2813 . . . . . . 7 (𝑥 = 𝐴 → (𝐹 ∈ (𝑈𝑚 𝑥) ↔ 𝐹 ∈ (𝑈𝑚 𝐴)))
1312imbi1d 330 . . . . . 6 (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈𝑚 𝑥) → ran 𝐹𝑈) ↔ (𝐹 ∈ (𝑈𝑚 𝐴) → ran 𝐹𝑈)))
1413rspccv 3434 . . . . 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 444 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → (𝐹 ∈ (𝑈𝑚 𝐴) → ran 𝐹𝑈))
171, 16sylbird 250 . 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 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  𝒫 cpw 4290  {cpr 4311   cuni 4576  Tr wtr 4892  ran crn 5255  wf 6033  (class class class)co 6801  𝑚 cmap 8011  Univcgru 9775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-tr 4893  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-map 8013  df-gru 9776
This theorem is referenced by:  gruiun  9784  grurn  9786  intgru  9799
  Copyright terms: Public domain W3C validator