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Theorem setsidvald 17076
Description: Value of the structure replacement function, deduction version.

Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (πΈβ€˜ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.)

Hypotheses
Ref Expression
setsidvald.e 𝐸 = Slot 𝑁
setsidvald.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
setsidvald.f (πœ‘ β†’ Fun 𝑆)
setsidvald.d (πœ‘ β†’ 𝑁 ∈ dom 𝑆)
Assertion
Ref Expression
setsidvald (πœ‘ β†’ 𝑆 = (𝑆 sSet βŸ¨π‘, (πΈβ€˜π‘†)⟩))

Proof of Theorem setsidvald
StepHypRef Expression
1 setsidvald.s . . 3 (πœ‘ β†’ 𝑆 ∈ 𝑉)
2 fvex 6856 . . 3 (πΈβ€˜π‘†) ∈ V
3 setsval 17044 . . 3 ((𝑆 ∈ 𝑉 ∧ (πΈβ€˜π‘†) ∈ V) β†’ (𝑆 sSet βŸ¨π‘, (πΈβ€˜π‘†)⟩) = ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (πΈβ€˜π‘†)⟩}))
41, 2, 3sylancl 587 . 2 (πœ‘ β†’ (𝑆 sSet βŸ¨π‘, (πΈβ€˜π‘†)⟩) = ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (πΈβ€˜π‘†)⟩}))
5 setsidvald.e . . . . . 6 𝐸 = Slot 𝑁
65, 1strfvnd 17062 . . . . 5 (πœ‘ β†’ (πΈβ€˜π‘†) = (π‘†β€˜π‘))
76opeq2d 4838 . . . 4 (πœ‘ β†’ βŸ¨π‘, (πΈβ€˜π‘†)⟩ = βŸ¨π‘, (π‘†β€˜π‘)⟩)
87sneqd 4599 . . 3 (πœ‘ β†’ {βŸ¨π‘, (πΈβ€˜π‘†)⟩} = {βŸ¨π‘, (π‘†β€˜π‘)⟩})
98uneq2d 4124 . 2 (πœ‘ β†’ ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (πΈβ€˜π‘†)⟩}) = ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (π‘†β€˜π‘)⟩}))
10 setsidvald.f . . 3 (πœ‘ β†’ Fun 𝑆)
11 setsidvald.d . . 3 (πœ‘ β†’ 𝑁 ∈ dom 𝑆)
12 funresdfunsn 7136 . . 3 ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) β†’ ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (π‘†β€˜π‘)⟩}) = 𝑆)
1310, 11, 12syl2anc 585 . 2 (πœ‘ β†’ ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (π‘†β€˜π‘)⟩}) = 𝑆)
144, 9, 133eqtrrd 2778 1 (πœ‘ β†’ 𝑆 = (𝑆 sSet βŸ¨π‘, (πΈβ€˜π‘†)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βˆ– cdif 3908   βˆͺ cun 3909  {csn 4587  βŸ¨cop 4593  dom cdm 5634   β†Ύ cres 5636  Fun wfun 6491  β€˜cfv 6497  (class class class)co 7358   sSet csts 17040  Slot cslot 17058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-sets 17041  df-slot 17059
This theorem is referenced by:  ressval3d  17132
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