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Theorem nfvres 5321
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 5010 . . . . . . . . . 10 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
2 df-iota 4967 . . . . . . . . . 10 (℩𝑥𝐴(𝐹𝐵)𝑥) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
31, 2eqtri 2108 . . . . . . . . 9 ((𝐹𝐵)‘𝐴) = {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}
43eleq2i 2154 . . . . . . . 8 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ 𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
5 eluni 3651 . . . . . . . 8 (𝑧 {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
64, 5bitri 182 . . . . . . 7 (𝑧 ∈ ((𝐹𝐵)‘𝐴) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}))
7 exsimpr 1554 . . . . . . 7 (∃𝑤(𝑧𝑤𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}}) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
86, 7sylbi 119 . . . . . 6 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}})
9 df-clab 2075 . . . . . . . 8 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦})
10 nfv 1466 . . . . . . . . 9 𝑦{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}
11 sneq 3452 . . . . . . . . . 10 (𝑦 = 𝑤 → {𝑦} = {𝑤})
1211eqeq2d 2099 . . . . . . . . 9 (𝑦 = 𝑤 → ({𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤}))
1310, 12sbie 1721 . . . . . . . 8 ([𝑤 / 𝑦]{𝑥𝐴(𝐹𝐵)𝑥} = {𝑦} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
149, 13bitri 182 . . . . . . 7 (𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1514exbii 1541 . . . . . 6 (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥𝐴(𝐹𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
168, 15sylib 120 . . . . 5 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
17 euabsn2 3506 . . . . 5 (∃!𝑥 𝐴(𝐹𝐵)𝑥 ↔ ∃𝑤{𝑥𝐴(𝐹𝐵)𝑥} = {𝑤})
1816, 17sylibr 132 . . . 4 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹𝐵)𝑥)
19 euex 1978 . . . 4 (∃!𝑥 𝐴(𝐹𝐵)𝑥 → ∃𝑥 𝐴(𝐹𝐵)𝑥)
20 df-br 3838 . . . . . . . 8 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵))
21 df-res 4440 . . . . . . . . 9 (𝐹𝐵) = (𝐹 ∩ (𝐵 × V))
2221eleq2i 2154 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
2320, 22bitri 182 . . . . . . 7 (𝐴(𝐹𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24 elin 3181 . . . . . . . 8 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
2524simprbi 269 . . . . . . 7 (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
2623, 25sylbi 119 . . . . . 6 (𝐴(𝐹𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27 opelxp1 4460 . . . . . 6 (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴𝐵)
2826, 27syl 14 . . . . 5 (𝐴(𝐹𝐵)𝑥𝐴𝐵)
2928exlimiv 1534 . . . 4 (∃𝑥 𝐴(𝐹𝐵)𝑥𝐴𝐵)
3018, 19, 293syl 17 . . 3 (𝑧 ∈ ((𝐹𝐵)‘𝐴) → 𝐴𝐵)
3130con3i 597 . 2 𝐴𝐵 → ¬ 𝑧 ∈ ((𝐹𝐵)‘𝐴))
3231eq0rdv 3324 1 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wex 1426  wcel 1438  [wsb 1692  ∃!weu 1948  {cab 2074  Vcvv 2619  cin 2996  c0 3284  {csn 3441  cop 3444   cuni 3648   class class class wbr 3837   × cxp 4426  cres 4430  cio 4965  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-res 4440  df-iota 4967  df-fv 5010
This theorem is referenced by: (None)
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