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Theorem bj-restb 37095
Description: An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restb (𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))

Proof of Theorem bj-restb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . 8 (𝐴𝐵𝐴𝐵)
2 ssidd 4007 . . . . . . . 8 (𝐴𝐵𝐴𝐴)
31, 2ssind 4241 . . . . . . 7 (𝐴𝐵𝐴 ⊆ (𝐵𝐴))
4 inss2 4238 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐴
54a1i 11 . . . . . . 7 (𝐴𝐵 → (𝐵𝐴) ⊆ 𝐴)
63, 5eqssd 4001 . . . . . 6 (𝐴𝐵𝐴 = (𝐵𝐴))
7 eleq1 2829 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
8 ineq1 4213 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦𝐴) = (𝐵𝐴))
98eqeq2d 2748 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 = (𝑦𝐴) ↔ 𝐴 = (𝐵𝐴)))
107, 9anbi12d 632 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑦𝑋𝐴 = (𝑦𝐴)) ↔ (𝐵𝑋𝐴 = (𝐵𝐴))))
1110spcegv 3597 . . . . . . . 8 (𝐵𝑋 → ((𝐵𝑋𝐴 = (𝐵𝐴)) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
1211expd 415 . . . . . . 7 (𝐵𝑋 → (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))))
1312pm2.43i 52 . . . . . 6 (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
146, 13mpan9 506 . . . . 5 ((𝐴𝐵𝐵𝑋) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
15 df-rex 3071 . . . . 5 (∃𝑦𝑋 𝐴 = (𝑦𝐴) ↔ ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
1614, 15sylibr 234 . . . 4 ((𝐴𝐵𝐵𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
1716adantl 481 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
18 ssexg 5323 . . . 4 ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ V)
19 elrest 17472 . . . 4 ((𝑋𝑉𝐴 ∈ V) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2018, 19sylan2 593 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2117, 20mpbird 257 . 2 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → 𝐴 ∈ (𝑋t 𝐴))
2221ex 412 1 (𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by:  bj-restv  37096  bj-resta  37097
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