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Theorem intgru 9924
Description: The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
intgru ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)

Proof of Theorem intgru
Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 478 . . 3 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2 intex 5012 . . 3 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
31, 2sylib 210 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
4 dfss3 3787 . . . . 5 (𝐴 ⊆ Univ ↔ ∀𝑢𝐴 𝑢 ∈ Univ)
5 grutr 9903 . . . . . 6 (𝑢 ∈ Univ → Tr 𝑢)
65ralimi 3133 . . . . 5 (∀𝑢𝐴 𝑢 ∈ Univ → ∀𝑢𝐴 Tr 𝑢)
74, 6sylbi 209 . . . 4 (𝐴 ⊆ Univ → ∀𝑢𝐴 Tr 𝑢)
8 trint 4961 . . . 4 (∀𝑢𝐴 Tr 𝑢 → Tr 𝐴)
97, 8syl 17 . . 3 (𝐴 ⊆ Univ → Tr 𝐴)
109adantr 473 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr 𝐴)
11 grupw 9905 . . . . . . . . . 10 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → 𝒫 𝑥𝑢)
1211ex 402 . . . . . . . . 9 (𝑢 ∈ Univ → (𝑥𝑢 → 𝒫 𝑥𝑢))
1312ral2imi 3128 . . . . . . . 8 (∀𝑢𝐴 𝑢 ∈ Univ → (∀𝑢𝐴 𝑥𝑢 → ∀𝑢𝐴 𝒫 𝑥𝑢))
14 vex 3388 . . . . . . . . 9 𝑥 ∈ V
1514elint2 4674 . . . . . . . 8 (𝑥 𝐴 ↔ ∀𝑢𝐴 𝑥𝑢)
16 vpwex 5047 . . . . . . . . 9 𝒫 𝑥 ∈ V
1716elint2 4674 . . . . . . . 8 (𝒫 𝑥 𝐴 ↔ ∀𝑢𝐴 𝒫 𝑥𝑢)
1813, 15, 173imtr4g 288 . . . . . . 7 (∀𝑢𝐴 𝑢 ∈ Univ → (𝑥 𝐴 → 𝒫 𝑥 𝐴))
1918imp 396 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
2019adantlr 707 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
21 r19.26 3245 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) ↔ (∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢))
22 grupr 9907 . . . . . . . . . . . 12 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23223expia 1151 . . . . . . . . . . 11 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦𝑢 → {𝑥, 𝑦} ∈ 𝑢))
2423ral2imi 3128 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
2521, 24sylbir 227 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
26 vex 3388 . . . . . . . . . 10 𝑦 ∈ V
2726elint2 4674 . . . . . . . . 9 (𝑦 𝐴 ↔ ∀𝑢𝐴 𝑦𝑢)
28 prex 5100 . . . . . . . . . 10 {𝑥, 𝑦} ∈ V
2928elint2 4674 . . . . . . . . 9 ({𝑥, 𝑦} ∈ 𝐴 ↔ ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢)
3025, 27, 293imtr4g 288 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3115, 30sylan2b 588 . . . . . . 7 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3231ralrimiv 3146 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
3332adantlr 707 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
34 elmapg 8108 . . . . . . . . . 10 (( 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3534elvd 3390 . . . . . . . . 9 ( 𝐴 ∈ V → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
362, 35sylbi 209 . . . . . . . 8 (𝐴 ≠ ∅ → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3736ad2antlr 719 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
38 intss1 4682 . . . . . . . . . . . 12 (𝑢𝐴 𝐴𝑢)
39 fss 6269 . . . . . . . . . . . 12 ((𝑦:𝑥 𝐴 𝐴𝑢) → 𝑦:𝑥𝑢)
4038, 39sylan2 587 . . . . . . . . . . 11 ((𝑦:𝑥 𝐴𝑢𝐴) → 𝑦:𝑥𝑢)
4140ralrimiva 3147 . . . . . . . . . 10 (𝑦:𝑥 𝐴 → ∀𝑢𝐴 𝑦:𝑥𝑢)
42 gruurn 9908 . . . . . . . . . . . . . 14 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦:𝑥𝑢) → ran 𝑦𝑢)
43423expia 1151 . . . . . . . . . . . . 13 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦:𝑥𝑢 ran 𝑦𝑢))
4443ral2imi 3128 . . . . . . . . . . . 12 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4521, 44sylbir 227 . . . . . . . . . . 11 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4615, 45sylan2b 588 . . . . . . . . . 10 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4741, 46syl5 34 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 → ∀𝑢𝐴 ran 𝑦𝑢))
4826rnex 7335 . . . . . . . . . . 11 ran 𝑦 ∈ V
4948uniex 7187 . . . . . . . . . 10 ran 𝑦 ∈ V
5049elint2 4674 . . . . . . . . 9 ( ran 𝑦 𝐴 ↔ ∀𝑢𝐴 ran 𝑦𝑢)
5147, 50syl6ibr 244 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5251adantlr 707 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5337, 52sylbid 232 . . . . . 6 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) → ran 𝑦 𝐴))
5453ralrimiv 3146 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴)
5520, 33, 543jca 1159 . . . 4 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
5655ralrimiva 3147 . . 3 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
574, 56sylanb 577 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
58 elgrug 9902 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ Univ ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))))
5958biimpar 470 . 2 (( 𝐴 ∈ V ∧ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))) → 𝐴 ∈ Univ)
603, 10, 57, 59syl12anc 866 1 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  wne 2971  wral 3089  Vcvv 3385  wss 3769  c0 4115  𝒫 cpw 4349  {cpr 4370   cuni 4628   cint 4667  Tr wtr 4945  ran crn 5313  wf 6097  (class class class)co 6878  𝑚 cmap 8095  Univcgru 9900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iin 4713  df-br 4844  df-opab 4906  df-tr 4946  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-gru 9901
This theorem is referenced by: (None)
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