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Theorem setsvalg 17129
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 3489 . 2 (𝑆𝑉𝑆 ∈ V)
2 elex 3489 . 2 (𝐴𝑊𝐴 ∈ V)
3 resexg 6026 . . . . 5 (𝑆 ∈ V → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
43adantr 480 . . . 4 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 ↾ (V ∖ dom {𝐴})) ∈ V)
5 snex 5428 . . . 4 {𝐴} ∈ V
6 unexg 7746 . . . 4 (((𝑆 ↾ (V ∖ dom {𝐴})) ∈ V ∧ {𝐴} ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
74, 5, 6sylancl 585 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V)
8 simpl 482 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑠 = 𝑆)
9 simpr 484 . . . . . . . . 9 ((𝑠 = 𝑆𝑒 = 𝐴) → 𝑒 = 𝐴)
109sneqd 4637 . . . . . . . 8 ((𝑠 = 𝑆𝑒 = 𝐴) → {𝑒} = {𝐴})
1110dmeqd 5903 . . . . . . 7 ((𝑠 = 𝑆𝑒 = 𝐴) → dom {𝑒} = dom {𝐴})
1211difeq2d 4119 . . . . . 6 ((𝑠 = 𝑆𝑒 = 𝐴) → (V ∖ dom {𝑒}) = (V ∖ dom {𝐴}))
138, 12reseq12d 5981 . . . . 5 ((𝑠 = 𝑆𝑒 = 𝐴) → (𝑠 ↾ (V ∖ dom {𝑒})) = (𝑆 ↾ (V ∖ dom {𝐴})))
1413, 10uneq12d 4161 . . . 4 ((𝑠 = 𝑆𝑒 = 𝐴) → ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
15 df-sets 17127 . . . 4 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
1614, 15ovmpoga 7570 . . 3 ((𝑆 ∈ V ∧ 𝐴 ∈ V ∧ ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}) ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
177, 16mpd3an3 1459 . 2 ((𝑆 ∈ V ∧ 𝐴 ∈ V) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
181, 2, 17syl2an 595 1 ((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3470  cdif 3942  cun 3943  {csn 4625  dom cdm 5673  cres 5675  (class class class)co 7415   sSet csts 17126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-res 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-sets 17127
This theorem is referenced by:  setsval  17130  setsdm  17133  setsfun  17134  setsfun0  17135  wunsets  17140  setsres  17141
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