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Theorem setsidel 44797
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 5383 . . . 4 𝐴, 𝐵⟩ ∈ V
21snid 4603 . . 3 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
3 elun2 4116 . . 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 16866 . . . 4 ((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
96, 7, 8syl2anc 584 . . 3 (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
105, 9eqtrid 2792 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
114, 10eleqtrrd 2844 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  Vcvv 3431  cdif 3889  cun 3890  {csn 4567  cop 4573  cres 5592  (class class class)co 7271   sSet csts 16862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-res 5602  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-sets 16863
This theorem is referenced by: (None)
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