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Theorem intgru 9588
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 477 . . 3 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
2 intex 4785 . . 3 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
31, 2sylib 208 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ V)
4 dfss3 3577 . . . . 5 (𝐴 ⊆ Univ ↔ ∀𝑢𝐴 𝑢 ∈ Univ)
5 grutr 9567 . . . . . 6 (𝑢 ∈ Univ → Tr 𝑢)
65ralimi 2947 . . . . 5 (∀𝑢𝐴 𝑢 ∈ Univ → ∀𝑢𝐴 Tr 𝑢)
74, 6sylbi 207 . . . 4 (𝐴 ⊆ Univ → ∀𝑢𝐴 Tr 𝑢)
8 trint 4733 . . . 4 (∀𝑢𝐴 Tr 𝑢 → Tr 𝐴)
97, 8syl 17 . . 3 (𝐴 ⊆ Univ → Tr 𝐴)
109adantr 481 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → Tr 𝐴)
11 grupw 9569 . . . . . . . . . 10 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → 𝒫 𝑥𝑢)
1211ex 450 . . . . . . . . 9 (𝑢 ∈ Univ → (𝑥𝑢 → 𝒫 𝑥𝑢))
1312ral2imi 2942 . . . . . . . 8 (∀𝑢𝐴 𝑢 ∈ Univ → (∀𝑢𝐴 𝑥𝑢 → ∀𝑢𝐴 𝒫 𝑥𝑢))
14 vex 3192 . . . . . . . . 9 𝑥 ∈ V
1514elint2 4452 . . . . . . . 8 (𝑥 𝐴 ↔ ∀𝑢𝐴 𝑥𝑢)
16 vpwex 4814 . . . . . . . . 9 𝒫 𝑥 ∈ V
1716elint2 4452 . . . . . . . 8 (𝒫 𝑥 𝐴 ↔ ∀𝑢𝐴 𝒫 𝑥𝑢)
1813, 15, 173imtr4g 285 . . . . . . 7 (∀𝑢𝐴 𝑢 ∈ Univ → (𝑥 𝐴 → 𝒫 𝑥 𝐴))
1918imp 445 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
2019adantlr 750 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → 𝒫 𝑥 𝐴)
21 r19.26 3058 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) ↔ (∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢))
22 grupr 9571 . . . . . . . . . . . 12 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦𝑢) → {𝑥, 𝑦} ∈ 𝑢)
23223expia 1264 . . . . . . . . . . 11 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦𝑢 → {𝑥, 𝑦} ∈ 𝑢))
2423ral2imi 2942 . . . . . . . . . 10 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
2521, 24sylbir 225 . . . . . . . . 9 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦𝑢 → ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢))
26 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
2726elint2 4452 . . . . . . . . 9 (𝑦 𝐴 ↔ ∀𝑢𝐴 𝑦𝑢)
28 prex 4875 . . . . . . . . . 10 {𝑥, 𝑦} ∈ V
2928elint2 4452 . . . . . . . . 9 ({𝑥, 𝑦} ∈ 𝐴 ↔ ∀𝑢𝐴 {𝑥, 𝑦} ∈ 𝑢)
3025, 27, 293imtr4g 285 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3115, 30sylan2b 492 . . . . . . 7 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦 𝐴 → {𝑥, 𝑦} ∈ 𝐴))
3231ralrimiv 2960 . . . . . 6 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
3332adantlr 750 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴)
34 elmapg 7822 . . . . . . . . . 10 (( 𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3514, 34mpan2 706 . . . . . . . . 9 ( 𝐴 ∈ V → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
362, 35sylbi 207 . . . . . . . 8 (𝐴 ≠ ∅ → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
3736ad2antlr 762 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) ↔ 𝑦:𝑥 𝐴))
38 intss1 4462 . . . . . . . . . . . 12 (𝑢𝐴 𝐴𝑢)
39 fss 6018 . . . . . . . . . . . 12 ((𝑦:𝑥 𝐴 𝐴𝑢) → 𝑦:𝑥𝑢)
4038, 39sylan2 491 . . . . . . . . . . 11 ((𝑦:𝑥 𝐴𝑢𝐴) → 𝑦:𝑥𝑢)
4140ralrimiva 2961 . . . . . . . . . 10 (𝑦:𝑥 𝐴 → ∀𝑢𝐴 𝑦:𝑥𝑢)
42 gruurn 9572 . . . . . . . . . . . . . 14 ((𝑢 ∈ Univ ∧ 𝑥𝑢𝑦:𝑥𝑢) → ran 𝑦𝑢)
43423expia 1264 . . . . . . . . . . . . 13 ((𝑢 ∈ Univ ∧ 𝑥𝑢) → (𝑦:𝑥𝑢 ran 𝑦𝑢))
4443ral2imi 2942 . . . . . . . . . . . 12 (∀𝑢𝐴 (𝑢 ∈ Univ ∧ 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4521, 44sylbir 225 . . . . . . . . . . 11 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ ∀𝑢𝐴 𝑥𝑢) → (∀𝑢𝐴 𝑦:𝑥𝑢 → ∀𝑢𝐴 ran 𝑦𝑢))
4615, 45sylan2b 492 . . . . . . . . . 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 7054 . . . . . . . . . . 11 ran 𝑦 ∈ V
4948uniex 6913 . . . . . . . . . 10 ran 𝑦 ∈ V
5049elint2 4452 . . . . . . . . 9 ( ran 𝑦 𝐴 ↔ ∀𝑢𝐴 ran 𝑦𝑢)
5147, 50syl6ibr 242 . . . . . . . 8 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5251adantlr 750 . . . . . . 7 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦:𝑥 𝐴 ran 𝑦 𝐴))
5337, 52sylbid 230 . . . . . 6 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝑦 ∈ ( 𝐴𝑚 𝑥) → ran 𝑦 𝐴))
5453ralrimiv 2960 . . . . 5 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴)
5520, 33, 543jca 1240 . . . 4 (((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) ∧ 𝑥 𝐴) → (𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
5655ralrimiva 2961 . . 3 ((∀𝑢𝐴 𝑢 ∈ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
574, 56sylanb 489 . 2 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))
58 elgrug 9566 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ Univ ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴(𝒫 𝑥 𝐴 ∧ ∀𝑦 𝐴{𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦 ∈ ( 𝐴𝑚 𝑥) ran 𝑦 𝐴))))
5958biimpar 502 . 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 1321 1 ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  wral 2907  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135  {cpr 4155   cuni 4407   cint 4445  Tr wtr 4717  ran crn 5080  wf 5848  (class class class)co 6610  𝑚 cmap 7809  Univcgru 9564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-tr 4718  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-gru 9565
This theorem is referenced by: (None)
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