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

Theorem wfgru 10807
Description: The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
wfgru (𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)

Proof of Theorem wfgru
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr3 5270 . . 3 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
2 r1elssi 9796 . . 3 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
31, 2mprgbir 3069 . 2 Tr (𝑅1 “ On)
4 pwwf 9798 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
54biimpi 215 . . . 4 (𝑥 (𝑅1 “ On) → 𝒫 𝑥 (𝑅1 “ On))
6 prwf 9802 . . . . 5 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
76ralrimiva 3147 . . . 4 (𝑥 (𝑅1 “ On) → ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On))
8 frn 6721 . . . . . . 7 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
9 vex 3479 . . . . . . . . . 10 𝑦 ∈ V
109rnex 7898 . . . . . . . . 9 ran 𝑦 ∈ V
1110r1elss 9797 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
12 uniwf 9810 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
1311, 12bitr3i 277 . . . . . . 7 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
148, 13sylib 217 . . . . . 6 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
1514ax-gen 1798 . . . . 5 𝑦(𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
1615a1i 11 . . . 4 (𝑥 (𝑅1 “ On) → ∀𝑦(𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On)))
175, 7, 163jca 1129 . . 3 (𝑥 (𝑅1 “ On) → (𝒫 𝑥 (𝑅1 “ On) ∧ ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))))
1817rgen 3064 . 2 𝑥 (𝑅1 “ On)(𝒫 𝑥 (𝑅1 “ On) ∧ ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On)))
19 ingru 10806 . 2 ((Tr (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(𝒫 𝑥 (𝑅1 “ On) ∧ ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On)))) → (𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ))
203, 18, 19mp2an 691 1 (𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wal 1540  wcel 2107  wral 3062  cin 3946  wss 3947  𝒫 cpw 4601  {cpr 4629   cuni 4907  Tr wtr 5264  ran crn 5676  cima 5678  Oncon0 6361  wf 6536  𝑅1cr1 9753  Univcgru 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-map 8818  df-r1 9755  df-rank 9756  df-gru 10782
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator