ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfvres GIF version

Theorem nfvres 5568
Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
nfvres 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fv 5243 . . . . . . . . . 10 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
2 df-iota 5196 . . . . . . . . . 10 (℩𝑥𝐴(𝐹𝐵)𝑥) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
31, 2eqtri 2210 . . . . . . . . 9 ((𝐹𝐵)‘𝐴) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
43eleq2i 2256 . . . . . . . 8 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ 𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
5 eluni 3827 . . . . . . . 8 (𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
64, 5bitri 184 . . . . . . 7 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
7 exsimpr 1629 . . . . . . 7 (∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
86, 7sylbi 121 . . . . . 6 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
9 df-clab 2176 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦})
10 nfv 1539 . . . . . . . . 9 𝑦{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}
11 sneq 3618 . . . . . . . . . 10 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1211eqeq2d 2201 . . . . . . . . 9 (𝑦 = 𝑤 → ({𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}))
1310, 12sbie 1802 . . . . . . . 8 ([𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
149, 13bitri 184 . . . . . . 7 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1514exbii 1616 . . . . . 6 (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
168, 15sylib 122 . . . . 5 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
17 euabsn2 3676 . . . . 5 (∃!𝑥 𝐴(𝐹𝐵)𝑥 ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1816, 17sylibr 134 . . . 4 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹𝐵)𝑥)
19 euex 2068 . . . 4 (∃!𝑥 𝐴(𝐹𝐵)𝑥 → ∃𝑥 𝐴(𝐹𝐵)𝑥)
20 df-br 4019 . . . . . . . 8 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵))
21 df-res 4656 . . . . . . . . 9 (𝐹𝐵) = (𝐹 ∩ (𝐵 × V))
2221eleq2i 2256 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
2320, 22bitri 184 . . . . . . 7 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24 elin 3333 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
2524simprbi 275 . . . . . . 7 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
2623, 25sylbi 121 . . . . . 6 (𝐴(𝐹𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27 opelxp1 4678 . . . . . 6 (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴𝐵)
2826, 27syl 14 . . . . 5 (𝐴(𝐹𝐵)𝑥𝐴𝐵)
2928exlimiv 1609 . . . 4 (∃𝑥 𝐴(𝐹𝐵)𝑥𝐴𝐵)
3018, 19, 293syl 17 . . 3 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → 𝐴𝐵)
3130con3i 633 . 2 𝐴𝐵 → ¬ 𝑧 ∈ ((𝐹𝐵)‘𝐴))
3231eq0rdv 3482 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wex 1503  [wsb 1773  ∃!weu 2038  wcel 2160  {cab 2175  Vcvv 2752  cin 3143  c0 3437  {csn 3607  cop 3610   cuni 3824   class class class wbr 4018   × cxp 4642  cres 4646  cio 5194  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-res 4656  df-iota 5196  df-fv 5243
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator