Theorem wfgru 10091

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 5074	. . 3 ⊢ (Tr ∪ (𝑅1 “ On) ↔ ∀𝑥 ∈ ∪ (𝑅1 “ On)𝑥 ⊆ ∪ (𝑅1 “ On))
2		r1elssi 9087	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ⊆ ∪ (𝑅1 “ On))
3	1, 2	mprgbir 3122	. 2 ⊢ Tr ∪ (𝑅1 “ On)
4		pwwf 9089	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
5	4	biimpi 217	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
6		prwf 9093	. . . . 5 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
7	6	ralrimiva 3151	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On))
8		frn 6395	. . . . . . 7 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ran 𝑦 ⊆ ∪ (𝑅1 “ On))
9		vex 3443	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9	rnex 7480	. . . . . . . . 9 ⊢ ran 𝑦 ∈ V
11	10	r1elss 9088	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑦 ⊆ ∪ (𝑅1 “ On))
12		uniwf 9101	. . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
13	11, 12	bitr3i 278	. . . . . . 7 ⊢ (ran 𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
14	8, 13	sylib 219	. . . . . 6 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))
15	14	ax-gen 1781	. . . . 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 1121	. . 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 3117	. 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 10090	. 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 1080  ∀wal 1523   ∈ wcel 2083  ∀wral 3107   ∩ cin 3864   ⊆ wss 3865  𝒫 cpw 4459  {cpr 4480  ∪ cuni 4751  Tr wtr 5070  ran crn 5451   “ cima 5453  Oncon0 6073  ⟶wf 6228  𝑅₁cr1 9044  Univcgru 10065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-map 8265  df-r1 9046  df-rank 9047  df-gru 10066

This theorem is referenced by: (None)