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Theorem restuni3 45123
Description: The underlying set of a subspace induced by the subspace operator t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
restuni3.1 (𝜑𝐴𝑉)
restuni3.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
restuni3 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Proof of Theorem restuni3
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4911 . . . . . . . 8 (𝑥 (𝐴t 𝐵) ↔ ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
21biimpi 216 . . . . . . 7 (𝑥 (𝐴t 𝐵) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
32adantl 481 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
4 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → 𝑦 ∈ (𝐴t 𝐵))
5 restuni3.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
6 restuni3.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
7 elrest 17472 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
85, 6, 7syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
98adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
104, 9mpbid 232 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
11103adant3 1133 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
12 simpl 482 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥𝑦)
13 simpr 484 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑦 = (𝑧𝐵))
1412, 13eleqtrd 2843 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥 ∈ (𝑧𝐵))
1514ex 412 . . . . . . . . . . . 12 (𝑥𝑦 → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
16153ad2ant3 1136 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
1716reximdv 3170 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (∃𝑧𝐴 𝑦 = (𝑧𝐵) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
1811, 17mpd 15 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
19183exp 1120 . . . . . . . 8 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) → (𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))))
2019rexlimdv 3153 . . . . . . 7 (𝜑 → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
2120adantr 480 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
223, 21mpd 15 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
23 elinel1 4201 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝐵) → 𝑥𝑧)
2423adantl 481 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝑧)
25 simpl 482 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑧𝐴)
26 elunii 4912 . . . . . . . . . 10 ((𝑥𝑧𝑧𝐴) → 𝑥 𝐴)
2724, 25, 26syl2anc 584 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 𝐴)
28 elinel2 4202 . . . . . . . . . 10 (𝑥 ∈ (𝑧𝐵) → 𝑥𝐵)
2928adantl 481 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝐵)
3027, 29elind 4200 . . . . . . . 8 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3130ex 412 . . . . . . 7 (𝑧𝐴 → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3231adantl 481 . . . . . 6 (((𝜑𝑥 (𝐴t 𝐵)) ∧ 𝑧𝐴) → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3332rexlimdva 3155 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑧𝐴 𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3422, 33mpd 15 . . . 4 ((𝜑𝑥 (𝐴t 𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3534ralrimiva 3146 . . 3 (𝜑 → ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
36 dfss3 3972 . . 3 ( (𝐴t 𝐵) ⊆ ( 𝐴𝐵) ↔ ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
3735, 36sylibr 234 . 2 (𝜑 (𝐴t 𝐵) ⊆ ( 𝐴𝐵))
38 elinel1 4201 . . . . . 6 (𝑥 ∈ ( 𝐴𝐵) → 𝑥 𝐴)
39 eluni2 4911 . . . . . 6 (𝑥 𝐴 ↔ ∃𝑧𝐴 𝑥𝑧)
4038, 39sylib 218 . . . . 5 (𝑥 ∈ ( 𝐴𝐵) → ∃𝑧𝐴 𝑥𝑧)
4140adantl 481 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑧𝐴 𝑥𝑧)
425adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐴𝑉)
436adantr 480 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐵𝑊)
44 simpr 484 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝑧𝐴)
45 eqid 2737 . . . . . . . . . 10 (𝑧𝐵) = (𝑧𝐵)
4642, 43, 44, 45elrestd 45113 . . . . . . . . 9 ((𝜑𝑧𝐴) → (𝑧𝐵) ∈ (𝐴t 𝐵))
47463adant3 1133 . . . . . . . 8 ((𝜑𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
48473adant1r 1178 . . . . . . 7 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
49 simp3 1139 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝑧)
50 simp1r 1199 . . . . . . . . 9 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ ( 𝐴𝐵))
51 elinel2 4202 . . . . . . . . 9 (𝑥 ∈ ( 𝐴𝐵) → 𝑥𝐵)
5250, 51syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝐵)
53 simpl 482 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝑧)
54 simpr 484 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝐵)
5553, 54elind 4200 . . . . . . . 8 ((𝑥𝑧𝑥𝐵) → 𝑥 ∈ (𝑧𝐵))
5649, 52, 55syl2anc 584 . . . . . . 7 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ (𝑧𝐵))
57 eleq2 2830 . . . . . . . 8 (𝑦 = (𝑧𝐵) → (𝑥𝑦𝑥 ∈ (𝑧𝐵)))
5857rspcev 3622 . . . . . . 7 (((𝑧𝐵) ∈ (𝐴t 𝐵) ∧ 𝑥 ∈ (𝑧𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
5948, 56, 58syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
60593exp 1120 . . . . 5 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (𝑧𝐴 → (𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)))
6160rexlimdv 3153 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (∃𝑧𝐴 𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦))
6241, 61mpd 15 . . 3 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
6362, 1sylibr 234 . 2 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → 𝑥 (𝐴t 𝐵))
6437, 63eqelssd 4005 1 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951   cuni 4907  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by:  restuni4  45126  subsalsal  46374
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