Theorem nfvres 5609

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 5278	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥)
2		df-iota 5231	. . . . . . . . . 10 ⊢ (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
3	1, 2	eqtri 2225	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
4	3	eleq2i 2271	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ 𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
5		eluni 3852	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
6	4, 5	bitri 184	. . . . . . 7 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
7		exsimpr 1640	. . . . . . 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 121	. . . . . 6 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
9		df-clab 2191	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦})
10		nfv 1550	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}
11		sneq 3643	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
12	11	eqeq2d 2216	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}))
13	10, 12	sbie 1813	. . . . . . . 8 ⊢ ([𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
14	9, 13	bitri 184	. . . . . . 7 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
15	14	exbii 1627	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
16	8, 15	sylib 122	. . . . 5 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
17		euabsn2 3701	. . . . 5 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
18	16, 17	sylibr 134	. . . 4 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
19		euex 2083	. . . 4 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → ∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
20		df-br 4044	. . . . . . . 8 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵))
21		df-res 4686	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝐵) = (𝐹 ∩ (𝐵 × V))
22	21	eleq2i 2271	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
23	20, 22	bitri 184	. . . . . . 7 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24		elin 3355	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
25	24	simprbi 275	. . . . . . 7 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
26	23, 25	sylbi 121	. . . . . 6 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27		opelxp1 4708	. . . . . 6 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴 ∈ 𝐵)
28	26, 27	syl 14	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
29	28	exlimiv 1620	. . . 4 ⊢ (∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
30	18, 19, 29	3syl 17	. . 3 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → 𝐴 ∈ 𝐵)
31	30	con3i 633	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴))
32	31	eq0rdv 3504	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1372  ∃wex 1514  [wsb 1784  ∃!weu 2053   ∈ wcel 2175  {cab 2190  Vcvv 2771   ∩ cin 3164  ∅c0 3459  {csn 3632  ⟨cop 3635  ∪ cuni 3849   class class class wbr 4043   × cxp 4672   ↾ cres 4676  ℩cio 5229  ‘cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4680  df-res 4686  df-iota 5231  df-fv 5278

This theorem is referenced by: (None)