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Theorem restuni3 39818
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 4592 . . . . . . . 8 (𝑥 (𝐴t 𝐵) ↔ ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
21biimpi 206 . . . . . . 7 (𝑥 (𝐴t 𝐵) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
32adantl 473 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
4 simpr 479 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → 𝑦 ∈ (𝐴t 𝐵))
5 restuni3.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
6 restuni3.2 . . . . . . . . . . . . . 14 (𝜑𝐵𝑊)
7 elrest 16310 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
85, 6, 7syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
98adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → (𝑦 ∈ (𝐴t 𝐵) ↔ ∃𝑧𝐴 𝑦 = (𝑧𝐵)))
104, 9mpbid 222 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵)) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
11103adant3 1127 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑦 = (𝑧𝐵))
12 simpl 474 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥𝑦)
13 simpr 479 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑦 = (𝑧𝐵))
1412, 13eleqtrd 2841 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐵)) → 𝑥 ∈ (𝑧𝐵))
1514ex 449 . . . . . . . . . . . 12 (𝑥𝑦 → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
16153ad2ant3 1130 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (𝑦 = (𝑧𝐵) → 𝑥 ∈ (𝑧𝐵)))
1716reximdv 3154 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → (∃𝑧𝐴 𝑦 = (𝑧𝐵) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
1811, 17mpd 15 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐴t 𝐵) ∧ 𝑥𝑦) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
19183exp 1113 . . . . . . . 8 (𝜑 → (𝑦 ∈ (𝐴t 𝐵) → (𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))))
2019rexlimdv 3168 . . . . . . 7 (𝜑 → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
2120adantr 472 . . . . . 6 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦 → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵)))
223, 21mpd 15 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → ∃𝑧𝐴 𝑥 ∈ (𝑧𝐵))
23 elinel1 3942 . . . . . . . . . . 11 (𝑥 ∈ (𝑧𝐵) → 𝑥𝑧)
2423adantl 473 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝑧)
25 simpl 474 . . . . . . . . . 10 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑧𝐴)
26 elunii 4593 . . . . . . . . . 10 ((𝑥𝑧𝑧𝐴) → 𝑥 𝐴)
2724, 25, 26syl2anc 696 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 𝐴)
28 elinel2 3943 . . . . . . . . . 10 (𝑥 ∈ (𝑧𝐵) → 𝑥𝐵)
2928adantl 473 . . . . . . . . 9 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥𝐵)
3027, 29elind 3941 . . . . . . . 8 ((𝑧𝐴𝑥 ∈ (𝑧𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3130ex 449 . . . . . . 7 (𝑧𝐴 → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3231adantl 473 . . . . . 6 (((𝜑𝑥 (𝐴t 𝐵)) ∧ 𝑧𝐴) → (𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3332rexlimdva 3169 . . . . 5 ((𝜑𝑥 (𝐴t 𝐵)) → (∃𝑧𝐴 𝑥 ∈ (𝑧𝐵) → 𝑥 ∈ ( 𝐴𝐵)))
3422, 33mpd 15 . . . 4 ((𝜑𝑥 (𝐴t 𝐵)) → 𝑥 ∈ ( 𝐴𝐵))
3534ralrimiva 3104 . . 3 (𝜑 → ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
36 dfss3 3733 . . 3 ( (𝐴t 𝐵) ⊆ ( 𝐴𝐵) ↔ ∀𝑥 (𝐴t 𝐵)𝑥 ∈ ( 𝐴𝐵))
3735, 36sylibr 224 . 2 (𝜑 (𝐴t 𝐵) ⊆ ( 𝐴𝐵))
38 elinel1 3942 . . . . . . . 8 (𝑥 ∈ ( 𝐴𝐵) → 𝑥 𝐴)
39 eluni2 4592 . . . . . . . 8 (𝑥 𝐴 ↔ ∃𝑧𝐴 𝑥𝑧)
4038, 39sylib 208 . . . . . . 7 (𝑥 ∈ ( 𝐴𝐵) → ∃𝑧𝐴 𝑥𝑧)
4140adantl 473 . . . . . 6 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑧𝐴 𝑥𝑧)
425adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → 𝐴𝑉)
436adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → 𝐵𝑊)
44 simpr 479 . . . . . . . . . . . 12 ((𝜑𝑧𝐴) → 𝑧𝐴)
45 eqid 2760 . . . . . . . . . . . 12 (𝑧𝐵) = (𝑧𝐵)
4642, 43, 44, 45elrestd 39808 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝑧𝐵) ∈ (𝐴t 𝐵))
47463adant3 1127 . . . . . . . . . 10 ((𝜑𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
48473adant1r 1188 . . . . . . . . 9 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → (𝑧𝐵) ∈ (𝐴t 𝐵))
49 simp3 1133 . . . . . . . . . 10 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝑧)
50 simp1r 1241 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ ( 𝐴𝐵))
51 elinel2 3943 . . . . . . . . . . 11 (𝑥 ∈ ( 𝐴𝐵) → 𝑥𝐵)
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥𝐵)
53 simpl 474 . . . . . . . . . . 11 ((𝑥𝑧𝑥𝐵) → 𝑥𝑧)
54 simpr 479 . . . . . . . . . . 11 ((𝑥𝑧𝑥𝐵) → 𝑥𝐵)
5553, 54elind 3941 . . . . . . . . . 10 ((𝑥𝑧𝑥𝐵) → 𝑥 ∈ (𝑧𝐵))
5649, 52, 55syl2anc 696 . . . . . . . . 9 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → 𝑥 ∈ (𝑧𝐵))
57 eleq2 2828 . . . . . . . . . 10 (𝑦 = (𝑧𝐵) → (𝑥𝑦𝑥 ∈ (𝑧𝐵)))
5857rspcev 3449 . . . . . . . . 9 (((𝑧𝐵) ∈ (𝐴t 𝐵) ∧ 𝑥 ∈ (𝑧𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
5948, 56, 58syl2anc 696 . . . . . . . 8 (((𝜑𝑥 ∈ ( 𝐴𝐵)) ∧ 𝑧𝐴𝑥𝑧) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
60593exp 1113 . . . . . . 7 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (𝑧𝐴 → (𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)))
6160rexlimdv 3168 . . . . . 6 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → (∃𝑧𝐴 𝑥𝑧 → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦))
6241, 61mpd 15 . . . . 5 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → ∃𝑦 ∈ (𝐴t 𝐵)𝑥𝑦)
6362, 1sylibr 224 . . . 4 ((𝜑𝑥 ∈ ( 𝐴𝐵)) → 𝑥 (𝐴t 𝐵))
6463ralrimiva 3104 . . 3 (𝜑 → ∀𝑥 ∈ ( 𝐴𝐵)𝑥 (𝐴t 𝐵))
65 dfss3 3733 . . 3 (( 𝐴𝐵) ⊆ (𝐴t 𝐵) ↔ ∀𝑥 ∈ ( 𝐴𝐵)𝑥 (𝐴t 𝐵))
6664, 65sylibr 224 . 2 (𝜑 → ( 𝐴𝐵) ⊆ (𝐴t 𝐵))
6737, 66eqssd 3761 1 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cin 3714  wss 3715   cuni 4588  (class class class)co 6814  t crest 16303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-rest 16305
This theorem is referenced by:  restuni4  39821  subsalsal  41098
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