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Theorem fvsetsid 15884
Description: The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
Assertion
Ref Expression
fvsetsid ((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)

Proof of Theorem fvsetsid
StepHypRef Expression
1 setsval 15882 . . . 4 ((𝐹𝑉𝑌𝑈) → (𝐹 sSet ⟨𝑋, 𝑌⟩) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
213adant2 1079 . . 3 ((𝐹𝑉𝑋𝑊𝑌𝑈) → (𝐹 sSet ⟨𝑋, 𝑌⟩) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
32fveq1d 6191 . 2 ((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩})‘𝑋))
4 simp2 1061 . . 3 ((𝐹𝑉𝑋𝑊𝑌𝑈) → 𝑋𝑊)
5 simp3 1062 . . 3 ((𝐹𝑉𝑋𝑊𝑌𝑈) → 𝑌𝑈)
6 neldifsn 4319 . . . . 5 ¬ 𝑋 ∈ (V ∖ {𝑋})
7 dmres 5417 . . . . . . 7 dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹)
8 inss1 3831 . . . . . . 7 ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋})
97, 8eqsstri 3633 . . . . . 6 dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋})
109sseli 3597 . . . . 5 (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋}))
116, 10mto 188 . . . 4 ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))
1211a1i 11 . . 3 ((𝐹𝑉𝑋𝑊𝑌𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})))
13 fsnunfv 6450 . . 3 ((𝑋𝑊𝑌𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
144, 5, 12, 13syl3anc 1325 . 2 ((𝐹𝑉𝑋𝑊𝑌𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)
153, 14eqtrd 2655 1 ((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1037   = wceq 1482  wcel 1989  Vcvv 3198  cdif 3569  cun 3570  cin 3571  {csn 4175  cop 4181  dom cdm 5112  cres 5114  cfv 5886  (class class class)co 6647   sSet csts 15849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-res 5124  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-sets 15858
This theorem is referenced by:  mdetunilem9  20420
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