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Theorem nfvres 5461
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 5138 . . . . . . . . . 10 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
2 df-iota 5095 . . . . . . . . . 10 (℩𝑥𝐴(𝐹𝐵)𝑥) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
31, 2eqtri 2161 . . . . . . . . 9 ((𝐹𝐵)‘𝐴) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
43eleq2i 2207 . . . . . . . 8 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ 𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
5 eluni 3746 . . . . . . . 8 (𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
64, 5bitri 183 . . . . . . 7 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
7 exsimpr 1598 . . . . . . 7 (∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
86, 7sylbi 120 . . . . . 6 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
9 df-clab 2127 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦})
10 nfv 1509 . . . . . . . . 9 𝑦{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}
11 sneq 3542 . . . . . . . . . 10 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1211eqeq2d 2152 . . . . . . . . 9 (𝑦 = 𝑤 → ({𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}))
1310, 12sbie 1765 . . . . . . . 8 ([𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
149, 13bitri 183 . . . . . . 7 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1514exbii 1585 . . . . . 6 (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
168, 15sylib 121 . . . . 5 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
17 euabsn2 3599 . . . . 5 (∃!𝑥 𝐴(𝐹𝐵)𝑥 ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1816, 17sylibr 133 . . . 4 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹𝐵)𝑥)
19 euex 2030 . . . 4 (∃!𝑥 𝐴(𝐹𝐵)𝑥 → ∃𝑥 𝐴(𝐹𝐵)𝑥)
20 df-br 3937 . . . . . . . 8 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵))
21 df-res 4558 . . . . . . . . 9 (𝐹𝐵) = (𝐹 ∩ (𝐵 × V))
2221eleq2i 2207 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
2320, 22bitri 183 . . . . . . 7 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24 elin 3263 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
2524simprbi 273 . . . . . . 7 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
2623, 25sylbi 120 . . . . . 6 (𝐴(𝐹𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27 opelxp1 4580 . . . . . 6 (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴𝐵)
2826, 27syl 14 . . . . 5 (𝐴(𝐹𝐵)𝑥𝐴𝐵)
2928exlimiv 1578 . . . 4 (∃𝑥 𝐴(𝐹𝐵)𝑥𝐴𝐵)
3018, 19, 293syl 17 . . 3 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → 𝐴𝐵)
3130con3i 622 . 2 𝐴𝐵 → ¬ 𝑧 ∈ ((𝐹𝐵)‘𝐴))
3231eq0rdv 3411 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  [wsb 1736  ∃!weu 2000  {cab 2126  Vcvv 2689  cin 3074  c0 3367  {csn 3531  cop 3534   cuni 3743   class class class wbr 3936   × cxp 4544  cres 4548  cio 5093  cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-xp 4552  df-res 4558  df-iota 5095  df-fv 5138
This theorem is referenced by: (None)
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