Theorem wfgru 10854

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 5271	. . 3 ⊢ (Tr ∪ (𝑅1 “ On) ↔ ∀𝑥 ∈ ∪ (𝑅1 “ On)𝑥 ⊆ ∪ (𝑅1 “ On))
2		r1elssi 9843	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ⊆ ∪ (𝑅1 “ On))
3	1, 2	mprgbir 3066	. 2 ⊢ Tr ∪ (𝑅1 “ On)
4		pwwf 9845	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
5	4	biimpi 216	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
6		prwf 9849	. . . . 5 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
7	6	ralrimiva 3144	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
8		frn 6744	. . . . . . 7 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ran 𝑦 ⊆ ∪ (𝑅1 “ On))
9		vex 3482	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	rnex 7933	. . . . . . . . 9 ⊢ ran 𝑦 ∈ V
11	10	r1elss 9844	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑦 ⊆ ∪ (𝑅1 “ On))
12		uniwf 9857	. . . . . . . 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 1792	. . . . 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 1127	. . 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 3061	. 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 10853	. 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 692	1 ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)

Syntax hints:   → wi 4   ∧ w3a 1086  ∀wal 1535   ∈ wcel 2106  ∀wral 3059   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  {cpr 4633  ∪ cuni 4912  Tr wtr 5265  ran crn 5690   “ cima 5692  Oncon0 6386  ⟶wf 6559  𝑅₁cr1 9800  Univcgru 10828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-r1 9802  df-rank 9803  df-gru 10829

This theorem is referenced by: (None)