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Theorem restuni6 38830
 Description: The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
restuni6.1 (𝜑𝐴𝑉)
restuni6.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
restuni6 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))

Proof of Theorem restuni6
StepHypRef Expression
1 restuni6.1 . . . 4 (𝜑𝐴𝑉)
2 restuni6.2 . . . 4 (𝜑𝐵𝑊)
3 eqid 2621 . . . . 5 𝐴 = 𝐴
43restin 20910 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴t 𝐵) = (𝐴t (𝐵 𝐴)))
51, 2, 4syl2anc 692 . . 3 (𝜑 → (𝐴t 𝐵) = (𝐴t (𝐵 𝐴)))
65unieqd 4419 . 2 (𝜑 (𝐴t 𝐵) = (𝐴t (𝐵 𝐴)))
7 inss2 3818 . . . 4 (𝐵 𝐴) ⊆ 𝐴
87a1i 11 . . 3 (𝜑 → (𝐵 𝐴) ⊆ 𝐴)
91, 8restuni4 38829 . 2 (𝜑 (𝐴t (𝐵 𝐴)) = (𝐵 𝐴))
10 incom 3789 . . 3 (𝐵 𝐴) = ( 𝐴𝐵)
1110a1i 11 . 2 (𝜑 → (𝐵 𝐴) = ( 𝐴𝐵))
126, 9, 113eqtrd 2659 1 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987   ∩ cin 3559   ⊆ wss 3560  ∪ cuni 4409  (class class class)co 6615   ↾t crest 16021 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-rest 16023 This theorem is referenced by:  unirestss  38832
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