Theorem wfgru 10556

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.

Step	Hyp	Ref	Expression
1		dftr3 5199	. . 3 ⊢ (Tr ∪ (𝑅1 “ On) ↔ ∀𝑥 ∈ ∪ (𝑅1 “ On)𝑥 ⊆ ∪ (𝑅1 “ On))
2		r1elssi 9547	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ⊆ ∪ (𝑅1 “ On))
3	1, 2	mprgbir 3080	. 2 ⊢ Tr ∪ (𝑅1 “ On)
4		pwwf 9549	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
5	4	biimpi 215	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
6		prwf 9553	. . . . 5 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
7	6	ralrimiva 3109	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
8		frn 6603	. . . . . . 7 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ran 𝑦 ⊆ ∪ (𝑅1 “ On))
9		vex 3434	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	rnex 7746	. . . . . . . . 9 ⊢ ran 𝑦 ∈ V
11	10	r1elss 9548	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑦 ⊆ ∪ (𝑅1 “ On))
12		uniwf 9561	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
13	11, 12	bitr3i 276	. . . . . . 7 ⊢ (ran 𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
14	8, 13	sylib 217	. . . . . 6 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
15	14	ax-gen 1801	. . . . 5 ⊢ ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
16	15	a1i 11	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)))
17	5, 7, 16	3jca 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))))
18	17	rgen 3075	. 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 10555	. 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))
20	3, 18, 19	mp2an 688	1 ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)

Syntax hints:   → wi 4   ∧ w3a 1085  ∀wal 1539   ∈ wcel 2109  ∀wral 3065   ∩ cin 3890   ⊆ wss 3891  𝒫 cpw 4538  {cpr 4568  ∪ cuni 4844  Tr wtr 5195  ran crn 5589   “ cima 5591  Oncon0 6263  ⟶wf 6426  𝑅₁cr1 9504  Univcgru 10530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-map 8591  df-r1 9506  df-rank 9507  df-gru 10531

This theorem is referenced by: (None)