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Theorem elrestd 41529
Description: A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
elrestd.1 (𝜑𝐽𝑉)
elrestd.2 (𝜑𝐵𝑊)
elrestd.3 (𝜑𝑋𝐽)
elrestd.4 𝐴 = (𝑋𝐵)
Assertion
Ref Expression
elrestd (𝜑𝐴 ∈ (𝐽t 𝐵))

Proof of Theorem elrestd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elrestd.3 . . 3 (𝜑𝑋𝐽)
2 elrestd.4 . . . 4 𝐴 = (𝑋𝐵)
32a1i 11 . . 3 (𝜑𝐴 = (𝑋𝐵))
4 ineq1 4159 . . . 4 (𝑥 = 𝑋 → (𝑥𝐵) = (𝑋𝐵))
54rspceeqv 3617 . . 3 ((𝑋𝐽𝐴 = (𝑋𝐵)) → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
61, 3, 5syl2anc 586 . 2 (𝜑 → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
7 elrestd.1 . . 3 (𝜑𝐽𝑉)
8 elrestd.2 . . 3 (𝜑𝐵𝑊)
9 elrest 16680 . . 3 ((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
107, 8, 9syl2anc 586 . 2 (𝜑 → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
116, 10mpbird 259 1 (𝜑𝐴 ∈ (𝐽t 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  wrex 3126  cin 3912  (class class class)co 7133  t crest 16673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-rest 16675
This theorem is referenced by:  restuni3  41538  subsaliuncl  42789  subsalsal  42790  sssmf  43163  mbfresmf  43164  smfconst  43174  smflimlem1  43195  smfres  43213  smfco  43225  smfsuplem1  43233
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