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Theorem wfgru 9676
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 4789 . . 3 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
2 r1elssi 8706 . . 3 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
31, 2mprgbir 2956 . 2 Tr (𝑅1 “ On)
4 pwwf 8708 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
54biimpi 206 . . . 4 (𝑥 (𝑅1 “ On) → 𝒫 𝑥 (𝑅1 “ On))
6 prwf 8712 . . . . 5 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
76ralrimiva 2995 . . . 4 (𝑥 (𝑅1 “ On) → ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On))
8 frn 6091 . . . . . . 7 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
9 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
109rnex 7142 . . . . . . . . 9 ran 𝑦 ∈ V
1110r1elss 8707 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
12 uniwf 8720 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
1311, 12bitr3i 266 . . . . . . 7 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
148, 13sylib 208 . . . . . 6 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
1514ax-gen 1762 . . . . 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 1261 . . 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 2951 . 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 9675 . 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 708 1 (𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054  wal 1521  wcel 2030  wral 2941  cin 3606  wss 3607  𝒫 cpw 4191  {cpr 4212   cuni 4468  Tr wtr 4785  ran crn 5144  cima 5146  Oncon0 5761  wf 5922  𝑅1cr1 8663  Univcgru 9650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-map 7901  df-r1 8665  df-rank 8666  df-gru 9651
This theorem is referenced by: (None)
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