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Theorem restuni3 43807
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 4913 . . . . . . . 8 (𝑥 (𝐴t 𝐵) ↔ ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
21biimpi 215 . . . . . . 7 (𝑥 (𝐴t 𝐵) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
32adantl 483 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
4 simpr 486 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → 𝑦 ∈ (𝐴t 𝐵))
5 restuni3.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
6 restuni3.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
7 elrest 17373 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
85, 6, 7syl2anc 585 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
98adantr 482 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
104, 9mpbid 231 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
11103adant3 1133 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
12 simpl 484 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥𝑦)
13 simpr 486 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑦 = (𝑧𝐵))
1412, 13eleqtrd 2836 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥 ∈ (𝑧𝐵))
1514ex 414 . . . . . . . . . . . 12 (𝑥𝑦 → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
16153ad2ant3 1136 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
1716reximdv 3171 . . . . . . . . . 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 3154 . . . . . . 7 (𝜑 → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
2120adantr 482 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
223, 21mpd 15 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
23 elinel1 4196 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝐵) → 𝑥𝑧)
2423adantl 483 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝑧)
25 simpl 484 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑧𝐴)
26 elunii 4914 . . . . . . . . . 10 ((𝑥𝑧𝑧𝐴) → 𝑥 𝐴)
2724, 25, 26syl2anc 585 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 𝐴)
28 elinel2 4197 . . . . . . . . . 10 (𝑥 ∈ (𝑧𝐵) → 𝑥𝐵)
2928adantl 483 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝐵)
3027, 29elind 4195 . . . . . . . 8 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3130ex 414 . . . . . . 7 (𝑧𝐴 → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3231adantl 483 . . . . . 6 (((𝜑𝑥 (𝐴t 𝐵)) ∧ 𝑧𝐴) → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3332rexlimdva 3156 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑧𝐴 𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3422, 33mpd 15 . . . 4 ((𝜑𝑥 (𝐴t 𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3534ralrimiva 3147 . . 3 (𝜑 → ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
36 dfss3 3971 . . 3 ( (𝐴t 𝐵) ⊆ ( 𝐴𝐵) ↔ ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
3735, 36sylibr 233 . 2 (𝜑 (𝐴t 𝐵) ⊆ ( 𝐴𝐵))
38 elinel1 4196 . . . . . 6 (𝑥 ∈ ( 𝐴𝐵) → 𝑥 𝐴)
39 eluni2 4913 . . . . . 6 (𝑥 𝐴 ↔ ∃𝑧𝐴 𝑥𝑧)
4038, 39sylib 217 . . . . 5 (𝑥 ∈ ( 𝐴𝐵) → ∃𝑧𝐴 𝑥𝑧)
4140adantl 483 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑧𝐴 𝑥𝑧)
425adantr 482 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐴𝑉)
436adantr 482 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐵𝑊)
44 simpr 486 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝑧𝐴)
45 eqid 2733 . . . . . . . . . 10 (𝑧𝐵) = (𝑧𝐵)
4642, 43, 44, 45elrestd 43797 . . . . . . . . 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 4197 . . . . . . . . 9 (𝑥 ∈ ( 𝐴𝐵) → 𝑥𝐵)
5250, 51syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝐵)
53 simpl 484 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝑧)
54 simpr 486 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝐵)
5553, 54elind 4195 . . . . . . . 8 ((𝑥𝑧𝑥𝐵) → 𝑥 ∈ (𝑧𝐵))
5649, 52, 55syl2anc 585 . . . . . . 7 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ (𝑧𝐵))
57 eleq2 2823 . . . . . . . 8 (𝑦 = (𝑧𝐵) → (𝑥𝑦𝑥 ∈ (𝑧𝐵)))
5857rspcev 3613 . . . . . . 7 (((𝑧𝐵) ∈ (𝐴t 𝐵) ∧ 𝑥 ∈ (𝑧𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
5948, 56, 58syl2anc 585 . . . . . 6 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
60593exp 1120 . . . . 5 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (𝑧𝐴 → (𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)))
6160rexlimdv 3154 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (∃𝑧𝐴 𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦))
6241, 61mpd 15 . . 3 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
6362, 1sylibr 233 . 2 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → 𝑥 (𝐴t 𝐵))
6437, 63eqelssd 4004 1 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cin 3948  wss 3949   cuni 4909  (class class class)co 7409  t crest 17366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rest 17368
This theorem is referenced by:  restuni4  43810  subsalsal  45075
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