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Theorem wfgru 10266
 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 5140 . . 3 (Tr (𝑅1 “ On) ↔ ∀𝑥 (𝑅1 “ On)𝑥 (𝑅1 “ On))
2 r1elssi 9257 . . 3 (𝑥 (𝑅1 “ On) → 𝑥 (𝑅1 “ On))
31, 2mprgbir 3086 . 2 Tr (𝑅1 “ On)
4 pwwf 9259 . . . . 5 (𝑥 (𝑅1 “ On) ↔ 𝒫 𝑥 (𝑅1 “ On))
54biimpi 219 . . . 4 (𝑥 (𝑅1 “ On) → 𝒫 𝑥 (𝑅1 “ On))
6 prwf 9263 . . . . 5 ((𝑥 (𝑅1 “ On) ∧ 𝑦 (𝑅1 “ On)) → {𝑥, 𝑦} ∈ (𝑅1 “ On))
76ralrimiva 3114 . . . 4 (𝑥 (𝑅1 “ On) → ∀𝑦 (𝑅1 “ On){𝑥, 𝑦} ∈ (𝑅1 “ On))
8 frn 6502 . . . . . . 7 (𝑦:𝑥 (𝑅1 “ On) → ran 𝑦 (𝑅1 “ On))
9 vex 3414 . . . . . . . . . 10 𝑦 ∈ V
109rnex 7620 . . . . . . . . 9 ran 𝑦 ∈ V
1110r1elss 9258 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
12 uniwf 9271 . . . . . . . 8 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
1311, 12bitr3i 280 . . . . . . 7 (ran 𝑦 (𝑅1 “ On) ↔ ran 𝑦 (𝑅1 “ On))
148, 13sylib 221 . . . . . 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 1126 . . 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 3081 . 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 10265 . 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 1085  ∀wal 1537   ∈ wcel 2112  ∀wral 3071   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4492  {cpr 4522  ∪ cuni 4796  Tr wtr 5136  ran crn 5523   “ cima 5525  Oncon0 6167  ⟶wf 6329  𝑅1cr1 9214  Univcgru 10240 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-map 8416  df-r1 9216  df-rank 9217  df-gru 10241 This theorem is referenced by: (None)
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