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Theorem bj-restb 34010
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 3915 . . . . . . . 8 (𝐴𝐵𝐴𝐴)
31, 2ssind 4133 . . . . . . 7 (𝐴𝐵𝐴 ⊆ (𝐵𝐴))
4 inss2 4130 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐴
54a1i 11 . . . . . . 7 (𝐴𝐵 → (𝐵𝐴) ⊆ 𝐴)
63, 5eqssd 3910 . . . . . 6 (𝐴𝐵𝐴 = (𝐵𝐴))
7 eleq1 2870 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
8 ineq1 4105 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦𝐴) = (𝐵𝐴))
98eqeq2d 2805 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 = (𝑦𝐴) ↔ 𝐴 = (𝐵𝐴)))
107, 9anbi12d 630 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑦𝑋𝐴 = (𝑦𝐴)) ↔ (𝐵𝑋𝐴 = (𝐵𝐴))))
1110spcegv 3540 . . . . . . . 8 (𝐵𝑋 → ((𝐵𝑋𝐴 = (𝐵𝐴)) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
1211expd 416 . . . . . . 7 (𝐵𝑋 → (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))))
1312pm2.43i 52 . . . . . 6 (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
146, 13mpan9 507 . . . . 5 ((𝐴𝐵𝐵𝑋) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
15 df-rex 3111 . . . . 5 (∃𝑦𝑋 𝐴 = (𝑦𝐴) ↔ ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
1614, 15sylibr 235 . . . 4 ((𝐴𝐵𝐵𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
1716adantl 482 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
18 ssexg 5123 . . . 4 ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ V)
19 elrest 16535 . . . 4 ((𝑋𝑉𝐴 ∈ V) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2018, 19sylan2 592 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2117, 20mpbird 258 . 2 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → 𝐴 ∈ (𝑋t 𝐴))
2221ex 413 1 (𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  wrex 3106  Vcvv 3437  cin 3862  wss 3863  (class class class)co 7021  t crest 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-rest 16530
This theorem is referenced by:  bj-restv  34011  bj-resta  34012
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