Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  restuni3 Structured version   Visualization version   GIF version

Theorem restuni3 40059
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 4632 . . . . . . . 8 (𝑥 (𝐴t 𝐵) ↔ ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
21biimpi 208 . . . . . . 7 (𝑥 (𝐴t 𝐵) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
32adantl 474 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
4 simpr 478 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → 𝑦 ∈ (𝐴t 𝐵))
5 restuni3.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
6 restuni3.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
7 elrest 16403 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
85, 6, 7syl2anc 580 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
98adantr 473 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
104, 9mpbid 224 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
11103adant3 1163 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
12 simpl 475 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥𝑦)
13 simpr 478 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑦 = (𝑧𝐵))
1412, 13eleqtrd 2880 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥 ∈ (𝑧𝐵))
1514ex 402 . . . . . . . . . . . 12 (𝑥𝑦 → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
16153ad2ant3 1166 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
1716reximdv 3196 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (∃𝑧𝐴 𝑦 = (𝑧𝐵) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
1811, 17mpd 15 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
19183exp 1149 . . . . . . . 8 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) → (𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))))
2019rexlimdv 3211 . . . . . . 7 (𝜑 → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
2120adantr 473 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
223, 21mpd 15 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
23 elinel1 3997 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝐵) → 𝑥𝑧)
2423adantl 474 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝑧)
25 simpl 475 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑧𝐴)
26 elunii 4633 . . . . . . . . . 10 ((𝑥𝑧𝑧𝐴) → 𝑥 𝐴)
2724, 25, 26syl2anc 580 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 𝐴)
28 elinel2 3998 . . . . . . . . . 10 (𝑥 ∈ (𝑧𝐵) → 𝑥𝐵)
2928adantl 474 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝐵)
3027, 29elind 3996 . . . . . . . 8 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3130ex 402 . . . . . . 7 (𝑧𝐴 → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3231adantl 474 . . . . . 6 (((𝜑𝑥 (𝐴t 𝐵)) ∧ 𝑧𝐴) → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3332rexlimdva 3212 . . . . 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 3787 . . 3 ( (𝐴t 𝐵) ⊆ ( 𝐴𝐵) ↔ ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
3735, 36sylibr 226 . 2 (𝜑 (𝐴t 𝐵) ⊆ ( 𝐴𝐵))
38 elinel1 3997 . . . . . 6 (𝑥 ∈ ( 𝐴𝐵) → 𝑥 𝐴)
39 eluni2 4632 . . . . . 6 (𝑥 𝐴 ↔ ∃𝑧𝐴 𝑥𝑧)
4038, 39sylib 210 . . . . 5 (𝑥 ∈ ( 𝐴𝐵) → ∃𝑧𝐴 𝑥𝑧)
4140adantl 474 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑧𝐴 𝑥𝑧)
425adantr 473 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐴𝑉)
436adantr 473 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝐵𝑊)
44 simpr 478 . . . . . . . . . 10 ((𝜑𝑧𝐴) → 𝑧𝐴)
45 eqid 2799 . . . . . . . . . 10 (𝑧𝐵) = (𝑧𝐵)
4642, 43, 44, 45elrestd 40049 . . . . . . . . 9 ((𝜑𝑧𝐴) → (𝑧𝐵) ∈ (𝐴t 𝐵))
47463adant3 1163 . . . . . . . 8 ((𝜑𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
48473adant1r 1224 . . . . . . 7 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
49 simp3 1169 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝑧)
50 simp1r 1256 . . . . . . . . 9 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ ( 𝐴𝐵))
51 elinel2 3998 . . . . . . . . 9 (𝑥 ∈ ( 𝐴𝐵) → 𝑥𝐵)
5250, 51syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝐵)
53 simpl 475 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝑧)
54 simpr 478 . . . . . . . . 9 ((𝑥𝑧𝑥𝐵) → 𝑥𝐵)
5553, 54elind 3996 . . . . . . . 8 ((𝑥𝑧𝑥𝐵) → 𝑥 ∈ (𝑧𝐵))
5649, 52, 55syl2anc 580 . . . . . . 7 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ (𝑧𝐵))
57 eleq2 2867 . . . . . . . 8 (𝑦 = (𝑧𝐵) → (𝑥𝑦𝑥 ∈ (𝑧𝐵)))
5857rspcev 3497 . . . . . . 7 (((𝑧𝐵) ∈ (𝐴t 𝐵) ∧ 𝑥 ∈ (𝑧𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
5948, 56, 58syl2anc 580 . . . . . 6 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
60593exp 1149 . . . . 5 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (𝑧𝐴 → (𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)))
6160rexlimdv 3211 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (∃𝑧𝐴 𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦))
6241, 61mpd 15 . . 3 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
6362, 1sylibr 226 . 2 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → 𝑥 (𝐴t 𝐵))
6437, 63eqelssd 3818 1 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wrex 3090  cin 3768  wss 3769   cuni 4628  (class class class)co 6878  t crest 16396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-rest 16398
This theorem is referenced by:  restuni4  40062  subsalsal  41320
  Copyright terms: Public domain W3C validator