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Theorem setsnidel 41117
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.s . . . 4 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆)
2 setsnidel.c . . . . . 6 (𝜑𝐶𝑋)
32elexd 3212 . . . . 5 (𝜑𝐶 ∈ V)
4 setsnidel.n . . . . . 6 (𝜑𝐴𝐶)
54necomd 2848 . . . . 5 (𝜑𝐶𝐴)
6 eldifsn 4315 . . . . 5 (𝐶 ∈ (V ∖ {𝐴}) ↔ (𝐶 ∈ V ∧ 𝐶𝐴))
73, 5, 6sylanbrc 698 . . . 4 (𝜑𝐶 ∈ (V ∖ {𝐴}))
8 setsnidel.d . . . . 5 (𝜑𝐷𝑌)
9 opelresg 5402 . . . . 5 (𝐷𝑌 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑆𝐶 ∈ (V ∖ {𝐴}))))
108, 9syl 17 . . . 4 (𝜑 → (⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑆𝐶 ∈ (V ∖ {𝐴}))))
111, 7, 10mpbir2and 957 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑆 ↾ (V ∖ {𝐴})))
12 elun1 3778 . . 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 15882 . . . 4 ((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1815, 16, 17syl2anc 693 . . 3 (𝜑 → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
1914, 18syl5eq 2667 . 2 (𝜑𝑅 = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2013, 19eleqtrrd 2703 1 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wne 2793  Vcvv 3198  cdif 3569  cun 3570  {csn 4175  cop 4181  cres 5114  (class class class)co 6647   sSet csts 15849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-res 5124  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-sets 15858
This theorem is referenced by: (None)
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