Theorem bj-restb 37128

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	⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴)))

Proof of Theorem bj-restb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2		ssidd 3953	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴)
3	1, 2	ssind 4186	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∩ 𝐴))
4		inss2 4183	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∩ 𝐴) ⊆ 𝐴)
6	3, 5	eqssd 3947	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐵 ∩ 𝐴))
7		eleq1 2819	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
8		ineq1 4158	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝐴) = (𝐵 ∩ 𝐴))
9	8	eqeq2d 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 = (𝑦 ∩ 𝐴) ↔ 𝐴 = (𝐵 ∩ 𝐴)))
10	7, 9	anbi12d 632	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴))))
11	10	spcegv 3547	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → ((𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))))
12	11	expd 415	. . . . . . 7 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 = (𝐵 ∩ 𝐴) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))))
13	12	pm2.43i 52	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → (𝐴 = (𝐵 ∩ 𝐴) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))))
14	6, 13	mpan9 506	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))
15		df-rex 3057	. . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))
16	14, 15	sylibr 234	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))
17	16	adantl 481	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))
18		ssexg 5256	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V)
19		elrest 17326	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)))
20	18, 19	sylan2 593	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)))
21	17, 20	mpbird 257	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ (𝑋 ↾t 𝐴))
22	21	ex 412	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  (class class class)co 7341   ↾_t crest 17319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-rest 17321

This theorem is referenced by:  bj-restv  37129  bj-resta  37130