MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intgru Structured version   Visualization version   GIF version

Theorem intgru 10843
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 483 . . 3 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2 intex 5341 . . 3 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
31, 2sylib 217 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
4 dfss3 3968 . . . . 5 (𝐴 ⊆ Univ ↔ ∀𝑢𝐴 𝑢 ∈ Univ)
5 grutr 10822 . . . . . 6 (𝑢 ∈ Univ → Tr 𝑢)
65ralimi 3079 . . . . 5 (∀𝑢𝐴 𝑢 ∈ Univ → ∀𝑢𝐴 Tr 𝑢)
74, 6sylbi 216 . . . 4 (𝐴 ⊆ Univ → ∀𝑢𝐴 Tr 𝑢)
8 trint 5285 . . . 4 (∀𝑢𝐴 Tr 𝑢 → Tr 𝐴)
97, 8syl 17 . . 3 (𝐴 ⊆ Univ → Tr 𝐴)
109adantr 479 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr 𝐴)
11 grupw 10824 . . . . . . . . . 10 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → 𝒫 𝑥𝑢)
1211ex 411 . . . . . . . . 9 (𝑢 ∈ Univ → (𝑥𝑢 → 𝒫 𝑥𝑢))
1312ral2imi 3081 . . . . . . . 8 (∀𝑢𝐴 𝑢 ∈ Univ → (∀𝑢𝐴 𝑥𝑢 → ∀𝑢𝐴 𝒫 𝑥𝑢))
14 vex 3475 . . . . . . . . 9 𝑥 ∈ V
1514elint2 4958 . . . . . . . 8 (𝑥 𝐴 ↔ ∀𝑢𝐴 𝑥𝑢)
16 vpwex 5379 . . . . . . . . 9 𝒫 𝑥 ∈ V
1716elint2 4958 . . . . . . . 8 (𝒫 𝑥 𝐴 ↔ ∀𝑢𝐴 𝒫 𝑥𝑢)
1813, 15, 173imtr4g 295 . . . . . . 7 (∀𝑢𝐴 𝑢 ∈ Univ → (𝑥 𝐴 → 𝒫 𝑥 𝐴))
1918imp 405 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
2019adantlr 713 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
21 r19.26 3107 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) ↔ (∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢))
22 grupr 10826 . . . . . . . . . . . 12 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23223expia 1118 . . . . . . . . . . 11 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦𝑢 → {𝑥, 𝑦} ∈ 𝑢))
2423ral2imi 3081 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
2521, 24sylbir 234 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
26 vex 3475 . . . . . . . . . 10 𝑦 ∈ V
2726elint2 4958 . . . . . . . . 9 (𝑦 𝐴 ↔ ∀𝑢𝐴 𝑦𝑢)
28 prex 5436 . . . . . . . . . 10 {𝑥, 𝑦} ∈ V
2928elint2 4958 . . . . . . . . 9 ({𝑥, 𝑦} ∈ 𝐴 ↔ ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢)
3025, 27, 293imtr4g 295 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3115, 30sylan2b 592 . . . . . . 7 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3231ralrimiv 3141 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
3332adantlr 713 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
34 elmapg 8862 . . . . . . . . . 10 (( 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ ( 𝐴m 𝑥) ↔ 𝑦:𝑥 𝐴))
3534elvd 3478 . . . . . . . . 9 ( 𝐴 ∈ V → (𝑦 ∈ ( 𝐴m 𝑥) ↔ 𝑦:𝑥 𝐴))
362, 35sylbi 216 . . . . . . . 8 (𝐴 ≠ ∅ → (𝑦 ∈ ( 𝐴m 𝑥) ↔ 𝑦:𝑥 𝐴))
3736ad2antlr 725 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴m 𝑥) ↔ 𝑦:𝑥 𝐴))
38 intss1 4968 . . . . . . . . . . . 12 (𝑢𝐴 𝐴𝑢)
39 fss 6742 . . . . . . . . . . . 12 ((𝑦:𝑥 𝐴 𝐴𝑢) → 𝑦:𝑥𝑢)
4038, 39sylan2 591 . . . . . . . . . . 11 ((𝑦:𝑥 𝐴𝑢𝐴) → 𝑦:𝑥𝑢)
4140ralrimiva 3142 . . . . . . . . . 10 (𝑦:𝑥 𝐴 → ∀𝑢𝐴 𝑦:𝑥𝑢)
42 gruurn 10827 . . . . . . . . . . . . . 14 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦:𝑥𝑢) → ran 𝑦𝑢)
43423expia 1118 . . . . . . . . . . . . 13 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦:𝑥𝑢 ran 𝑦𝑢))
4443ral2imi 3081 . . . . . . . . . . . 12 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4521, 44sylbir 234 . . . . . . . . . . 11 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4615, 45sylan2b 592 . . . . . . . . . 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 7922 . . . . . . . . . . 11 ran 𝑦 ∈ V
4948uniex 7750 . . . . . . . . . 10 ran 𝑦 ∈ V
5049elint2 4958 . . . . . . . . 9 ( ran 𝑦 𝐴 ↔ ∀𝑢𝐴 ran 𝑦𝑢)
5147, 50imbitrrdi 251 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5251adantlr 713 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5337, 52sylbid 239 . . . . . 6 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴m 𝑥) → ran 𝑦 𝐴))
5453ralrimiv 3141 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 ∈ ( 𝐴m 𝑥) ran 𝑦 𝐴)
5520, 33, 543jca 1125 . . . 4 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴m 𝑥) ran 𝑦 𝐴))
5655ralrimiva 3142 . . 3 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴m 𝑥) ran 𝑦 𝐴))
574, 56sylanb 579 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴m 𝑥) ran 𝑦 𝐴))
58 elgrug 10821 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ Univ ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴m 𝑥) ran 𝑦 𝐴))))
5958biimpar 476 . 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 835 1 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wcel 2098  wne 2936  wral 3057  Vcvv 3471  wss 3947  c0 4324  𝒫 cpw 4604  {cpr 4632   cuni 4910   cint 4951  Tr wtr 5267  ran crn 5681  wf 6547  (class class class)co 7424  m cmap 8849  Univcgru 10819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iin 5001  df-br 5151  df-opab 5213  df-tr 5268  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-gru 10820
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator