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Theorem setsnidel 43414
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 3512 . . . . 5 (𝜑𝐶 ∈ V)
3 setsnidel.n . . . . . 6 (𝜑𝐴𝐶)
43necomd 3068 . . . . 5 (𝜑𝐶𝐴)
5 eldifsn 4711 . . . . 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 5852 . . . . 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 709 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12 elun1 4149 . . 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 16501 . . . 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, 18syl5eq 2865 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2013, 19eleqtrrd 2913 1 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  cdif 3930  cun 3931  {csn 4557  cop 4563  cres 5550  (class class class)co 7145   sSet csts 16469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-sets 16478
This theorem is referenced by: (None)
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