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Theorem setsvalg 15934
Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
setsvalg ((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))

Proof of Theorem setsvalg
Dummy variables 𝑒 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝑆𝑉𝑆 ∈ V)
2 elex 3243 . 2 (𝐴𝑊𝐴 ∈ V)
3 resexg 5477 . . . . 5 (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
43adantr 480 . . . 4 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
5 snex 4938 . . . 4 {𝐴} ∈ V
6 unexg 7001 . . . 4 (((𝑆 ↾ (V ∖ dom {𝐴})) ∈ V ∧ {𝐴} ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
74, 5, 6sylancl 695 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
8 simpl 472 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑠 = 𝑆)
9 simpr 476 . . . . . . . . 9 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑒 = 𝐴)
109sneqd 4222 . . . . . . . 8 ((𝑠 = 𝑆𝑒 = 𝐴) → {𝑒} = {𝐴})
1110dmeqd 5358 . . . . . . 7 ((𝑠 = 𝑆𝑒 = 𝐴) → dom {𝑒} = dom {𝐴})
1211difeq2d 3761 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴}))
138, 12reseq12d 5429 . . . . 5 ((𝑠 = 𝑆𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴})))
1413, 10uneq12d 3801 . . . 4 ((𝑠 = 𝑆𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
15 df-sets 15911 . . . 4 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
1614, 15ovmpt2ga 6832 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V ∧ ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
177, 16mpd3an3 1465 . 2 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
181, 2, 17syl2an 493 1 ((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  {csn 4210  dom cdm 5143  cres 5145  (class class class)co 6690   sSet csts 15902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-sets 15911
This theorem is referenced by:  setsval  15935  setsdm  15939  setsfun  15940  setsfun0  15941  wunsets  15947  setsres  15948
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