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Theorem bj-restb 35293
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 3946 . . . . . . . 8 (𝐴𝐵𝐴𝐴)
31, 2ssind 4169 . . . . . . 7 (𝐴𝐵𝐴 ⊆ (𝐵𝐴))
4 inss2 4166 . . . . . . . 8 (𝐵𝐴) ⊆ 𝐴
54a1i 11 . . . . . . 7 (𝐴𝐵 → (𝐵𝐴) ⊆ 𝐴)
63, 5eqssd 3940 . . . . . 6 (𝐴𝐵𝐴 = (𝐵𝐴))
7 eleq1 2821 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
8 ineq1 4142 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦𝐴) = (𝐵𝐴))
98eqeq2d 2744 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐴 = (𝑦𝐴) ↔ 𝐴 = (𝐵𝐴)))
107, 9anbi12d 630 . . . . . . . . 9 (𝑦 = 𝐵 → ((𝑦𝑋𝐴 = (𝑦𝐴)) ↔ (𝐵𝑋𝐴 = (𝐵𝐴))))
1110spcegv 3538 . . . . . . . 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 233 . . . 4 ((𝐴𝐵𝐵𝑋) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
1716adantl 481 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → ∃𝑦𝑋 𝐴 = (𝑦𝐴))
18 ssexg 5250 . . . 4 ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ V)
19 elrest 17166 . . . 4 ((𝑋𝑉𝐴 ∈ V) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2018, 19sylan2 592 . . 3 ((𝑋𝑉 ∧ (𝐴𝐵𝐵𝑋)) → (𝐴 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝐴 = (𝑦𝐴)))
2117, 20mpbird 256 . 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 205  wa 395   = wceq 1537  wex 1777  wcel 2101  wrex 3068  Vcvv 3434  cin 3888  wss 3889  (class class class)co 7295  t crest 17159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-rest 17161
This theorem is referenced by:  bj-restv  35294  bj-resta  35295
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