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Theorem setsvalg 17142
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 3492 . 2 (𝑆𝑉𝑆 ∈ V)
2 elex 3492 . 2 (𝐴𝑊𝐴 ∈ V)
3 resexg 6036 . . . . 5 (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
43adantr 479 . . . 4 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
5 snex 5437 . . . 4 {𝐴} ∈ V
6 unexg 7757 . . . 4 (((𝑆 ↾ (V ∖ dom {𝐴})) ∈ V ∧ {𝐴} ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
74, 5, 6sylancl 584 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
8 simpl 481 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑠 = 𝑆)
9 simpr 483 . . . . . . . . 9 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑒 = 𝐴)
109sneqd 4644 . . . . . . . 8 ((𝑠 = 𝑆𝑒 = 𝐴) → {𝑒} = {𝐴})
1110dmeqd 5912 . . . . . . 7 ((𝑠 = 𝑆𝑒 = 𝐴) → dom {𝑒} = dom {𝐴})
1211difeq2d 4122 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴}))
138, 12reseq12d 5990 . . . . 5 ((𝑠 = 𝑆𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴})))
1413, 10uneq12d 4165 . . . 4 ((𝑠 = 𝑆𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
15 df-sets 17140 . . . 4 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
1614, 15ovmpoga 7581 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V ∧ ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
177, 16mpd3an3 1458 . 2 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
181, 2, 17syl2an 594 1 ((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cdif 3946  cun 3947  {csn 4632  dom cdm 5682  cres 5684  (class class class)co 7426   sSet csts 17139
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-sets 17140
This theorem is referenced by:  setsval  17143  setsdm  17146  setsfun  17147  setsfun0  17148  wunsets  17153  setsres  17154
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