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Theorem elrestd 39806
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 3956 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐵) = (𝑋𝐵))
54eqeq2d 2780 . . . 4 (𝑥 = 𝑋 → (𝐴 = (𝑥𝐵) ↔ 𝐴 = (𝑋𝐵)))
65rspcev 3458 . . 3 ((𝑋𝐽𝐴 = (𝑋𝐵)) → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
71, 3, 6syl2anc 565 . 2 (𝜑 → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
8 elrestd.1 . . 3 (𝜑𝐽𝑉)
9 elrestd.2 . . 3 (𝜑𝐵𝑊)
10 elrest 16295 . . 3 ((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
118, 9, 10syl2anc 565 . 2 (𝜑 → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
127, 11mpbird 247 1 (𝜑𝐴 ∈ (𝐽t 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wcel 2144  wrex 3061  cin 3720  (class class class)co 6792  t crest 16288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-rest 16290
This theorem is referenced by:  restuni3  39816  subsaliuncl  41087  subsalsal  41088  sssmf  41461  mbfresmf  41462  smfconst  41472  smflimlem1  41493  smfres  41511  smfco  41523  smfsuplem1  41531
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