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Theorem bj-restb 37077
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 4019 . . . . . . . 8 (𝐴𝐵𝐴𝐴)
31, 2ssind 4249 . . . . . . 7 (𝐴𝐵𝐴 ⊆ (𝐵𝐴))
4 inss2 4246 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐴
54a1i 11 . . . . . . 7 (𝐴𝐵 → (𝐵𝐴) ⊆ 𝐴)
63, 5eqssd 4013 . . . . . 6 (𝐴𝐵𝐴 = (𝐵𝐴))
7 eleq1 2827 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
8 ineq1 4221 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦𝐴) = (𝐵𝐴))
98eqeq2d 2746 . . . . . . . . . 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 3069 . . . . 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 5329 . . . 4 ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ V)
19 elrest 17474 . . . 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 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  cin 3962  wss 3963  (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:  bj-restv  37078  bj-resta  37079
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