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Theorem setsnidel 43834
 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 3489 . . . . 5 (𝜑𝐶 ∈ V)
3 setsnidel.n . . . . . 6 (𝜑𝐴𝐶)
43necomd 3066 . . . . 5 (𝜑𝐶𝐴)
5 eldifsn 4693 . . . . 5 (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶𝐴))
62, 4, 5sylanbrc 586 . . . 4 (𝜑𝐶 ∈ (V ∖ {𝐴}))
7 setsnidel.s . . . 4 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
8 setsnidel.d . . . . 5 (𝜑𝐷𝑌)
9 opelres 5837 . . . . 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 712 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12 elun1 4127 . . 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 16504 . . . 4 ((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1815, 16, 17syl2anc 587 . . 3 (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1914, 18syl5eq 2869 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2013, 19eleqtrrd 2917 1 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  Vcvv 3469   ∖ cdif 3905   ∪ cun 3906  {csn 4539  ⟨cop 4545   ↾ cres 5534  (class class class)co 7140   sSet csts 16472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-sets 16481 This theorem is referenced by: (None)
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