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Theorem bj-restb 36578
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 4003 . . . . . . . 8 (𝐴𝐵𝐴𝐴)
31, 2ssind 4233 . . . . . . 7 (𝐴𝐵𝐴 ⊆ (𝐵𝐴))
4 inss2 4230 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐴
54a1i 11 . . . . . . 7 (𝐴𝐵 → (𝐵𝐴) ⊆ 𝐴)
63, 5eqssd 3997 . . . . . 6 (𝐴𝐵𝐴 = (𝐵𝐴))
7 eleq1 2816 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
8 ineq1 4205 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦𝐴) = (𝐵𝐴))
98eqeq2d 2738 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 = (𝑦𝐴) ↔ 𝐴 = (𝐵𝐴)))
107, 9anbi12d 630 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑦𝑋𝐴 = (𝑦𝐴)) ↔ (𝐵𝑋𝐴 = (𝐵𝐴))))
1110spcegv 3584 . . . . . . . 8 (𝐵𝑋 → ((𝐵𝑋𝐴 = (𝐵𝐴)) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
1211expd 414 . . . . . . 7 (𝐵𝑋 → (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))))
1312pm2.43i 52 . . . . . 6 (𝐵𝑋 → (𝐴 = (𝐵𝐴) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴))))
146, 13mpan9 505 . . . . 5 ((𝐴𝐵𝐵𝑋) → ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
15 df-rex 3067 . . . . 5 (∃𝑦𝑋 𝐴 = (𝑦𝐴) ↔ ∃𝑦(𝑦𝑋𝐴 = (𝑦𝐴)))
1614, 15sylibr 233 . . . 4 ((𝐴𝐵𝐵𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
1716adantl 480 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
18 ssexg 5325 . . . 4 ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ V)
19 elrest 17414 . . . 4 ((𝑋𝑉𝐴 ∈ V) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2018, 19sylan2 591 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2117, 20mpbird 256 . 2 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → 𝐴 ∈ (𝑋t 𝐴))
2221ex 411 1 (𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wrex 3066  Vcvv 3471  cin 3946  wss 3947  (class class class)co 7424  t crest 17407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-rest 17409
This theorem is referenced by:  bj-restv  36579  bj-resta  36580
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