Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-restuni2 Structured version   Visualization version   GIF version

Theorem bj-restuni2 37081
Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23186 and restuni2 23191. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restuni2 ((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Proof of Theorem bj-restuni2
StepHypRef Expression
1 uniexg 7759 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 ssexg 5329 . . . . 5 ((𝐴 𝑋 𝑋 ∈ V) → 𝐴 ∈ V)
31, 2sylan2 593 . . . 4 ((𝐴 𝑋𝑋𝑉) → 𝐴 ∈ V)
43ancoms 458 . . 3 ((𝑋𝑉𝐴 𝑋) → 𝐴 ∈ V)
5 bj-restuni 37080 . . 3 ((𝑋𝑉𝐴 ∈ V) → (𝑋t 𝐴) = ( 𝑋𝐴))
64, 5syldan 591 . 2 ((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = ( 𝑋𝐴))
7 inss2 4246 . . . . 5 ( 𝑋𝐴) ⊆ 𝐴
87a1i 11 . . . 4 (𝐴 𝑋 → ( 𝑋𝐴) ⊆ 𝐴)
9 id 22 . . . . 5 (𝐴 𝑋𝐴 𝑋)
10 ssidd 4019 . . . . 5 (𝐴 𝑋𝐴𝐴)
119, 10ssind 4249 . . . 4 (𝐴 𝑋𝐴 ⊆ ( 𝑋𝐴))
128, 11eqssd 4013 . . 3 (𝐴 𝑋 → ( 𝑋𝐴) = 𝐴)
1312adantl 481 . 2 ((𝑋𝑉𝐴 𝑋) → ( 𝑋𝐴) = 𝐴)
146, 13eqtrd 2775 1 ((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963   cuni 4912  (class class class)co 7431  t crest 17467
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-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-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-rest 17469
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator