Theorem wfgru 10885

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 5289	. . 3 ⊢ (Tr ∪ (𝑅1 “ On) ↔ ∀𝑥 ∈ ∪ (𝑅1 “ On)𝑥 ⊆ ∪ (𝑅1 “ On))
2		r1elssi 9874	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ⊆ ∪ (𝑅1 “ On))
3	1, 2	mprgbir 3074	. 2 ⊢ Tr ∪ (𝑅1 “ On)
4		pwwf 9876	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
5	4	biimpi 216	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
6		prwf 9880	. . . . 5 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
7	6	ralrimiva 3152	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
8		frn 6754	. . . . . . 7 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ran 𝑦 ⊆ ∪ (𝑅1 “ On))
9		vex 3492	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	rnex 7950	. . . . . . . . 9 ⊢ ran 𝑦 ∈ V
11	10	r1elss 9875	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑦 ⊆ ∪ (𝑅1 “ On))
12		uniwf 9888	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
13	11, 12	bitr3i 277	. . . . . . 7 ⊢ (ran 𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
14	8, 13	sylib 218	. . . . . 6 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
15	14	ax-gen 1793	. . . . 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 1128	. . 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 3069	. 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 10884	. 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 691	1 ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)

Syntax hints:   → wi 4   ∧ w3a 1087  ∀wal 1535   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  {cpr 4650  ∪ cuni 4931  Tr wtr 5283  ran crn 5701   “ cima 5703  Oncon0 6395  ⟶wf 6569  𝑅₁cr1 9831  Univcgru 10859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-map 8886  df-r1 9833  df-rank 9834  df-gru 10860

This theorem is referenced by: (None)