Theorem nfvres 5529

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 5206	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥)
2		df-iota 5160	. . . . . . . . . 10 ⊢ (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
3	1, 2	eqtri 2191	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
4	3	eleq2i 2237	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ 𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
5		eluni 3799	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
6	4, 5	bitri 183	. . . . . . 7 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
7		exsimpr 1611	. . . . . . 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 120	. . . . . 6 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
9		df-clab 2157	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦})
10		nfv 1521	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}
11		sneq 3594	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
12	11	eqeq2d 2182	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}))
13	10, 12	sbie 1784	. . . . . . . 8 ⊢ ([𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
14	9, 13	bitri 183	. . . . . . 7 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
15	14	exbii 1598	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
16	8, 15	sylib 121	. . . . 5 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
17		euabsn2 3652	. . . . 5 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
18	16, 17	sylibr 133	. . . 4 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
19		euex 2049	. . . 4 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → ∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
20		df-br 3990	. . . . . . . 8 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵))
21		df-res 4623	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝐵) = (𝐹 ∩ (𝐵 × V))
22	21	eleq2i 2237	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
23	20, 22	bitri 183	. . . . . . 7 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24		elin 3310	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
25	24	simprbi 273	. . . . . . 7 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
26	23, 25	sylbi 120	. . . . . 6 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27		opelxp1 4645	. . . . . 6 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴 ∈ 𝐵)
28	26, 27	syl 14	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
29	28	exlimiv 1591	. . . 4 ⊢ (∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
30	18, 19, 29	3syl 17	. . 3 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → 𝐴 ∈ 𝐵)
31	30	con3i 627	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴))
32	31	eq0rdv 3459	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485  [wsb 1755  ∃!weu 2019   ∈ wcel 2141  {cab 2156  Vcvv 2730   ∩ cin 3120  ∅c0 3414  {csn 3583  ⟨cop 3586  ∪ cuni 3796   class class class wbr 3989   × cxp 4609   ↾ cres 4613  ℩cio 5158  ‘cfv 5198

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-res 4623  df-iota 5160  df-fv 5206

This theorem is referenced by: (None)