Theorem nfvres 5227

Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
nfvres	⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 4930	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥)
2		df-iota 4887	. . . . . . . . . 10 ⊢ (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
3	1, 2	eqtri 2101	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
4	3	eleq2i 2145	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ 𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
5		eluni 3604	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
6	4, 5	bitri 182	. . . . . . 7 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
7		exsimpr 1549	. . . . . . 7 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
8	6, 7	sylbi 119	. . . . . 6 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
9		df-clab 2068	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦})
10		nfv 1461	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}
11		sneq 3409	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
12	11	eqeq2d 2092	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}))
13	10, 12	sbie 1714	. . . . . . . 8 ⊢ ([𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
14	9, 13	bitri 182	. . . . . . 7 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
15	14	exbii 1536	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
16	8, 15	sylib 120	. . . . 5 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
17		euabsn2 3461	. . . . 5 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
18	16, 17	sylibr 132	. . . 4 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
19		euex 1971	. . . 4 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → ∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
20		df-br 3786	. . . . . . . 8 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵))
21		df-res 4375	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝐵) = (𝐹 ∩ (𝐵 × V))
22	21	eleq2i 2145	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
23	20, 22	bitri 182	. . . . . . 7 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24		elin 3155	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
25	24	simprbi 269	. . . . . . 7 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
26	23, 25	sylbi 119	. . . . . 6 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27		opelxp1 4395	. . . . . 6 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴 ∈ 𝐵)
28	26, 27	syl 14	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
29	28	exlimiv 1529	. . . 4 ⊢ (∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
30	18, 19, 29	3syl 17	. . 3 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → 𝐴 ∈ 𝐵)
31	30	con3i 594	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴))
32	31	eq0rdv 3288	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  [wsb 1685  ∃!weu 1941  {cab 2067  Vcvv 2601   ∩ cin 2972  ∅c0 3251  {csn 3398  ⟨cop 3401  ∪ cuni 3601   class class class wbr 3785   × cxp 4361   ↾ cres 4365  ℩cio 4885  ‘cfv 4922

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-res 4375  df-iota 4887  df-fv 4930

This theorem is referenced by: (None)