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Theorem wfgru 10856
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 5265 . . 3 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
2 r1elssi 9845 . . 3 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
31, 2mprgbir 3068 . 2 Tr (𝑅1 “ On)
4 pwwf 9847 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
54biimpi 216 . . . 4 (𝑥 (𝑅1 “ On) → 𝒫 𝑥 (𝑅1 “ On))
6 prwf 9851 . . . . 5 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
76ralrimiva 3146 . . . 4 (𝑥 (𝑅1 “ On) → ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On))
8 frn 6743 . . . . . . 7 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
9 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
109rnex 7932 . . . . . . . . 9 ran 𝑦 ∈ V
1110r1elss 9846 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
12 uniwf 9859 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
1311, 12bitr3i 277 . . . . . . 7 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
148, 13sylib 218 . . . . . 6 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
1514ax-gen 1795 . . . . 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 3063 . 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 10855 . 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 692 1 (𝑈 ∈ Univ → (𝑈 (𝑅1 “ On)) ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wal 1538  wcel 2108  wral 3061  cin 3950  wss 3951  𝒫 cpw 4600  {cpr 4628   cuni 4907  Tr wtr 5259  ran crn 5686  cima 5688  Oncon0 6384  wf 6557  𝑅1cr1 9802  Univcgru 10830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-r1 9804  df-rank 9805  df-gru 10831
This theorem is referenced by: (None)
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