Theorem nfvres 5206

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 4910	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥)
2		df-iota 4867	. . . . . . . . . 10 ⊢ (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
3	1, 2	eqtri 2060	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐵)‘𝐴) = ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}
4	3	eleq2i 2104	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ 𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
5		eluni 3583	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
6	4, 5	bitri 173	. . . . . . 7 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}}))
7		exsimpr 1509	. . . . . . 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 114	. . . . . 6 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}})
9		df-clab 2027	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ [𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦})
10		nfv 1421	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}
11		sneq 3386	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
12	11	eqeq2d 2051	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤}))
13	10, 12	sbie 1674	. . . . . . . 8 ⊢ ([𝑤 / 𝑦]{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
14	9, 13	bitri 173	. . . . . . 7 ⊢ (𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
15	14	exbii 1496	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ {𝑦 ∣ {𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑦}} ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
16	8, 15	sylib 127	. . . . 5 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
17		euabsn2 3439	. . . . 5 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ∃𝑤{𝑥 ∣ 𝐴(𝐹 ↾ 𝐵)𝑥} = {𝑤})
18	16, 17	sylibr 137	. . . 4 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → ∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
19		euex 1930	. . . 4 ⊢ (∃!𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → ∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥)
20		df-br 3765	. . . . . . . 8 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵))
21		df-res 4357	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝐵) = (𝐹 ∩ (𝐵 × V))
22	21	eleq2i 2104	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ↾ 𝐵) ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
23	20, 22	bitri 173	. . . . . . 7 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)))
24		elin 3126	. . . . . . . 8 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V)))
25	24	simprbi 260	. . . . . . 7 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐹 ∩ (𝐵 × V)) → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
26	23, 25	sylbi 114	. . . . . 6 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → ⟨𝐴, 𝑥⟩ ∈ (𝐵 × V))
27		opelxp1 4377	. . . . . 6 ⊢ (⟨𝐴, 𝑥⟩ ∈ (𝐵 × V) → 𝐴 ∈ 𝐵)
28	26, 27	syl 14	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
29	28	exlimiv 1489	. . . 4 ⊢ (∃𝑥 𝐴(𝐹 ↾ 𝐵)𝑥 → 𝐴 ∈ 𝐵)
30	18, 19, 29	3syl 17	. . 3 ⊢ (𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴) → 𝐴 ∈ 𝐵)
31	30	con3i 562	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝑧 ∈ ((𝐹 ↾ 𝐵)‘𝐴))
32	31	eq0rdv 3261	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  [wsb 1645  ∃!weu 1900  {cab 2026  Vcvv 2557   ∩ cin 2916  ∅c0 3224  {csn 3375  ⟨cop 3378  ∪ cuni 3580   class class class wbr 3764   × cxp 4343   ↾ cres 4347  ℩cio 4865  ‘cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910

This theorem is referenced by: (None)