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Theorem setsnidOLD 17246
Description: Obsolete version of setsnid 17245 as of 7-Nov-2024. Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
setsid.e 𝐸 = Slot (𝐸‘ndx)
setsnid.n (𝐸‘ndx) ≠ 𝐷
Assertion
Ref Expression
setsnidOLD (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))

Proof of Theorem setsnidOLD
StepHypRef Expression
1 setsid.e . . . 4 𝐸 = Slot (𝐸‘ndx)
2 id 22 . . . 4 (𝑊 ∈ V → 𝑊 ∈ V)
31, 2strfvnd 17222 . . 3 (𝑊 ∈ V → (𝐸𝑊) = (𝑊‘(𝐸‘ndx)))
4 ovex 7464 . . . . 5 (𝑊 sSet ⟨𝐷, 𝐶⟩) ∈ V
54, 1strfvn 17223 . . . 4 (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)) = ((𝑊 sSet ⟨𝐷, 𝐶⟩)‘(𝐸‘ndx))
6 setsres 17215 . . . . . 6 (𝑊 ∈ V → ((𝑊 sSet ⟨𝐷, 𝐶⟩) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷})))
76fveq1d 6908 . . . . 5 (𝑊 ∈ V → (((𝑊 sSet ⟨𝐷, 𝐶⟩) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)))
8 fvex 6919 . . . . . . 7 (𝐸‘ndx) ∈ V
9 setsnid.n . . . . . . 7 (𝐸‘ndx) ≠ 𝐷
10 eldifsn 4786 . . . . . . 7 ((𝐸‘ndx) ∈ (V ∖ {𝐷}) ↔ ((𝐸‘ndx) ∈ V ∧ (𝐸‘ndx) ≠ 𝐷))
118, 9, 10mpbir2an 711 . . . . . 6 (𝐸‘ndx) ∈ (V ∖ {𝐷})
12 fvres 6925 . . . . . 6 ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → (((𝑊 sSet ⟨𝐷, 𝐶⟩) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet ⟨𝐷, 𝐶⟩)‘(𝐸‘ndx)))
1311, 12ax-mp 5 . . . . 5 (((𝑊 sSet ⟨𝐷, 𝐶⟩) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet ⟨𝐷, 𝐶⟩)‘(𝐸‘ndx))
14 fvres 6925 . . . . . 6 ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx)))
1511, 14ax-mp 5 . . . . 5 ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))
167, 13, 153eqtr3g 2800 . . . 4 (𝑊 ∈ V → ((𝑊 sSet ⟨𝐷, 𝐶⟩)‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx)))
175, 16eqtrid 2789 . . 3 (𝑊 ∈ V → (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)) = (𝑊‘(𝐸‘ndx)))
183, 17eqtr4d 2780 . 2 (𝑊 ∈ V → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))
191str0 17226 . . 3 ∅ = (𝐸‘∅)
20 fvprc 6898 . . 3 𝑊 ∈ V → (𝐸𝑊) = ∅)
21 reldmsets 17202 . . . . 5 Rel dom sSet
2221ovprc1 7470 . . . 4 𝑊 ∈ V → (𝑊 sSet ⟨𝐷, 𝐶⟩) = ∅)
2322fveq2d 6910 . . 3 𝑊 ∈ V → (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)) = (𝐸‘∅))
2419, 20, 233eqtr4a 2803 . 2 𝑊 ∈ V → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))
2518, 24pm2.61i 182 1 (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  c0 4333  {csn 4626  cop 4632  cres 5687  cfv 6561  (class class class)co 7431   sSet csts 17200  Slot cslot 17218  ndxcnx 17230
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-mpt 5226  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  df-slot 17219
This theorem is referenced by: (None)
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