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 32232
 Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20776 and restuni2 20781. (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 6853 . . . . 5 (𝑋𝑉 𝑋 ∈ V)
2 ssexg 4732 . . . . 5 ((𝐴 𝑋 𝑋 ∈ V) → 𝐴 ∈ V)
31, 2sylan2 490 . . . 4 ((𝐴 𝑋𝑋𝑉) → 𝐴 ∈ V)
43ancoms 468 . . 3 ((𝑋𝑉𝐴 𝑋) → 𝐴 ∈ V)
5 bj-restuni 32231 . . 3 ((𝑋𝑉𝐴 ∈ V) → (𝑋t 𝐴) = ( 𝑋𝐴))
64, 5syldan 486 . 2 ((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = ( 𝑋𝐴))
7 inss2 3796 . . . . 5 ( 𝑋𝐴) ⊆ 𝐴
87a1i 11 . . . 4 (𝐴 𝑋 → ( 𝑋𝐴) ⊆ 𝐴)
9 id 22 . . . . 5 (𝐴 𝑋𝐴 𝑋)
10 ssid 3587 . . . . . 6 𝐴𝐴
1110a1i 11 . . . . 5 (𝐴 𝑋𝐴𝐴)
129, 11ssind 3799 . . . 4 (𝐴 𝑋𝐴 ⊆ ( 𝑋𝐴))
138, 12eqssd 3585 . . 3 (𝐴 𝑋 → ( 𝑋𝐴) = 𝐴)
1413adantl 481 . 2 ((𝑋𝑉𝐴 𝑋) → ( 𝑋𝐴) = 𝐴)
156, 14eqtrd 2644 1 ((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  (class class class)co 6549   ↾t crest 15904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator