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Theorem setsidel 42388
Description: The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
Hypotheses
Ref Expression
setsidel.s (𝜑𝑆𝑉)
setsidel.b (𝜑𝐵𝑊)
setsidel.r 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)
Assertion
Ref Expression
setsidel (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Proof of Theorem setsidel
StepHypRef Expression
1 opex 5166 . . . 4 𝐴, 𝐵⟩ ∈ V
21snid 4430 . . 3 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
3 elun2 4004 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝐴, 𝐵⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
42, 3mp1i 13 . 2 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
5 setsidel.r . . 3 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)
6 setsidel.s . . . 4 (𝜑𝑆𝑉)
7 setsidel.b . . . 4 (𝜑𝐵𝑊)
8 setsval 16289 . . . 4 ((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
96, 7, 8syl2anc 579 . . 3 (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
105, 9syl5eq 2826 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
114, 10eleqtrrd 2862 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  Vcvv 3398  cdif 3789  cun 3790  {csn 4398  cop 4404  cres 5359  (class class class)co 6924   sSet csts 16257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-res 5369  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-sets 16266
This theorem is referenced by: (None)
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