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Theorem setsidvald 17141
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 6898 . . 3 (πΈβ€˜π‘†) ∈ V
3 setsval 17109 . . 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 585 . 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 17127 . . . . 5 (πœ‘ β†’ (πΈβ€˜π‘†) = (π‘†β€˜π‘))
76opeq2d 4875 . . . 4 (πœ‘ β†’ βŸ¨π‘, (πΈβ€˜π‘†)⟩ = βŸ¨π‘, (π‘†β€˜π‘)⟩)
87sneqd 4635 . . 3 (πœ‘ β†’ {βŸ¨π‘, (πΈβ€˜π‘†)⟩} = {βŸ¨π‘, (π‘†β€˜π‘)⟩})
98uneq2d 4158 . 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 7183 . . 3 ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) β†’ ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (π‘†β€˜π‘)⟩}) = 𝑆)
1310, 11, 12syl2anc 583 . 2 (πœ‘ β†’ ((𝑆 β†Ύ (V βˆ– {𝑁})) βˆͺ {βŸ¨π‘, (π‘†β€˜π‘)⟩}) = 𝑆)
144, 9, 133eqtrrd 2771 1 (πœ‘ β†’ 𝑆 = (𝑆 sSet βŸ¨π‘, (πΈβ€˜π‘†)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βˆ– cdif 3940   βˆͺ cun 3941  {csn 4623  βŸ¨cop 4629  dom cdm 5669   β†Ύ cres 5671  Fun wfun 6531  β€˜cfv 6537  (class class class)co 7405   sSet csts 17105  Slot cslot 17123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-sets 17106  df-slot 17124
This theorem is referenced by:  ressval3d  17200  opprabs  33102
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