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Theorem nfvres 5350
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 5036 . . . . . . . . . 10 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
2 df-iota 4993 . . . . . . . . . 10 (℩𝑥𝐴(𝐹𝐵)𝑥) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
31, 2eqtri 2109 . . . . . . . . 9 ((𝐹𝐵)‘𝐴) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
43eleq2i 2155 . . . . . . . 8 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ 𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
5 eluni 3662 . . . . . . . 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 1555 . . . . . . 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 2076 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦})
10 nfv 1467 . . . . . . . . 9 𝑦{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}
11 sneq 3461 . . . . . . . . . 10 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1211eqeq2d 2100 . . . . . . . . 9 (𝑦 = 𝑤 → ({𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}))
1310, 12sbie 1722 . . . . . . . 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 1542 . . . . . 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 3515 . . . . 5 (∃!𝑥 𝐴(𝐹𝐵)𝑥 ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1816, 17sylibr 133 . . . 4 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹𝐵)𝑥)
19 euex 1979 . . . 4 (∃!𝑥 𝐴(𝐹𝐵)𝑥 → ∃𝑥 𝐴(𝐹𝐵)𝑥)
20 df-br 3852 . . . . . . . 8 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵))
21 df-res 4464 . . . . . . . . 9 (𝐹𝐵) = (𝐹 ∩ (𝐵 × V))
2221eleq2i 2155 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
2320, 22bitri 183 . . . . . . 7 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24 elin 3184 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
2524simprbi 270 . . . . . . 7 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
2623, 25sylbi 120 . . . . . 6 (𝐴(𝐹𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27 opelxp1 4485 . . . . . 6 (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴𝐵)
2826, 27syl 14 . . . . 5 (𝐴(𝐹𝐵)𝑥𝐴𝐵)
2928exlimiv 1535 . . . 4 (∃𝑥 𝐴(𝐹𝐵)𝑥𝐴𝐵)
3018, 19, 293syl 17 . . 3 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → 𝐴𝐵)
3130con3i 598 . 2 𝐴𝐵 → ¬ 𝑧 ∈ ((𝐹𝐵)‘𝐴))
3231eq0rdv 3331 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1290  wex 1427  wcel 1439  [wsb 1693  ∃!weu 1949  {cab 2075  Vcvv 2620  cin 2999  c0 3287  {csn 3450  cop 3453   cuni 3659   class class class wbr 3851   × cxp 4450  cres 4454  cio 4991  cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-xp 4458  df-res 4464  df-iota 4993  df-fv 5036
This theorem is referenced by: (None)
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