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Theorem nfvres 5193
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 4897 . . . . . . . . . 10 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
2 df-iota 4854 . . . . . . . . . 10 (℩𝑥𝐴(𝐹𝐵)𝑥) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
31, 2eqtri 2060 . . . . . . . . 9 ((𝐹𝐵)‘𝐴) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
43eleq2i 2104 . . . . . . . 8 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ 𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
5 eluni 3580 . . . . . . . 8 (𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
64, 5bitri 173 . . . . . . 7 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
7 exsimpr 1509 . . . . . . 7 (∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
86, 7sylbi 114 . . . . . 6 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
9 df-clab 2027 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦})
10 nfv 1421 . . . . . . . . 9 𝑦{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}
11 sneq 3383 . . . . . . . . . 10 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1211eqeq2d 2051 . . . . . . . . 9 (𝑦 = 𝑤 → ({𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}))
1310, 12sbie 1674 . . . . . . . 8 ([𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
149, 13bitri 173 . . . . . . 7 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1514exbii 1496 . . . . . 6 (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
168, 15sylib 127 . . . . 5 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
17 euabsn2 3436 . . . . 5 (∃!𝑥 𝐴(𝐹𝐵)𝑥 ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1816, 17sylibr 137 . . . 4 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹𝐵)𝑥)
19 euex 1930 . . . 4 (∃!𝑥 𝐴(𝐹𝐵)𝑥 → ∃𝑥 𝐴(𝐹𝐵)𝑥)
20 df-br 3762 . . . . . . . 8 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵))
21 df-res 4344 . . . . . . . . 9 (𝐹𝐵) = (𝐹 ∩ (𝐵 × V))
2221eleq2i 2104 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
2320, 22bitri 173 . . . . . . 7 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24 elin 3123 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
2524simprbi 260 . . . . . . 7 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
2623, 25sylbi 114 . . . . . 6 (𝐴(𝐹𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27 opelxp1 4364 . . . . . 6 (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴𝐵)
2826, 27syl 14 . . . . 5 (𝐴(𝐹𝐵)𝑥𝐴𝐵)
2928exlimiv 1489 . . . 4 (∃𝑥 𝐴(𝐹𝐵)𝑥𝐴𝐵)
3018, 19, 293syl 17 . . 3 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → 𝐴𝐵)
3130con3i 562 . 2 𝐴𝐵 → ¬ 𝑧 ∈ ((𝐹𝐵)‘𝐴))
3231eq0rdv 3258 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  [wsb 1645  ∃!weu 1900  {cab 2026  Vcvv 2554  cin 2913  c0 3221  {csn 3372  cop 3375   cuni 3577   class class class wbr 3761   × cxp 4330  cres 4334  cio 4852  cfv 4889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-xp 4338  df-res 4344  df-iota 4854  df-fv 4897
This theorem is referenced by: (None)
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