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Theorem setsnidel 47364
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⟩)
setsnidel.c (𝜑𝐶𝑋)
setsnidel.d (𝜑𝐷𝑌)
setsnidel.s (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
setsnidel.n (𝜑𝐴𝐶)
Assertion
Ref Expression
setsnidel (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)

Proof of Theorem setsnidel
StepHypRef Expression
1 setsnidel.c . . . . . 6 (𝜑𝐶𝑋)
21elexd 3504 . . . . 5 (𝜑𝐶 ∈ V)
3 setsnidel.n . . . . . 6 (𝜑𝐴𝐶)
43necomd 2996 . . . . 5 (𝜑𝐶𝐴)
5 eldifsn 4786 . . . . 5 (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶𝐴))
62, 4, 5sylanbrc 583 . . . 4 (𝜑𝐶 ∈ (V ∖ {𝐴}))
7 setsnidel.s . . . 4 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
8 setsnidel.d . . . . 5 (𝜑𝐷𝑌)
9 opelres 6003 . . . . 5 (𝐷𝑌 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑆)))
108, 9syl 17 . . . 4 (𝜑 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (𝐶 ∈ (V ∖ {𝐴}) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑆)))
116, 7, 10mpbir2and 713 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12 elun1 4182 . . 3 (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) → ⟨𝐶, 𝐷⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1311, 12syl 17 . 2 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
14 setsidel.r . . 3 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)
15 setsidel.s . . . 4 (𝜑𝑆𝑉)
16 setsidel.b . . . 4 (𝜑𝐵𝑊)
17 setsval 17204 . . . 4 ((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1815, 16, 17syl2anc 584 . . 3 (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1914, 18eqtrid 2789 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2013, 19eleqtrrd 2844 1 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  cun 3949  {csn 4626  cop 4632  cres 5687  (class class class)co 7431   sSet csts 17200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201
This theorem is referenced by: (None)
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