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Theorem elrestd 40044
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 4006 . . . 4 (𝑥 = 𝑋 → (𝑥𝐵) = (𝑋𝐵))
54rspceeqv 3516 . . 3 ((𝑋𝐽𝐴 = (𝑋𝐵)) → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
61, 3, 5syl2anc 580 . 2 (𝜑 → ∃𝑥𝐽 𝐴 = (𝑥𝐵))
7 elrestd.1 . . 3 (𝜑𝐽𝑉)
8 elrestd.2 . . 3 (𝜑𝐵𝑊)
9 elrest 16402 . . 3 ((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
107, 8, 9syl2anc 580 . 2 (𝜑 → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))
116, 10mpbird 249 1 (𝜑𝐴 ∈ (𝐽t 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wrex 3091  cin 3769  (class class class)co 6879  t crest 16395
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 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
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 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-rest 16397
This theorem is referenced by:  restuni3  40054  subsaliuncl  41314  subsalsal  41315  sssmf  41688  mbfresmf  41689  smfconst  41699  smflimlem1  41720  smfres  41738  smfco  41750  smfsuplem1  41758
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