Theorem fvprc 2829

Description: A function's value at a proper class is the empty set.

Assertion

Ref	Expression
fvprc	⊢ (¬ A ∈ V → (F ‘A) = ∅)

Proof of Theorem fvprc

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ x ∈ V
2	1	snnz 1846	. . . . . . 7 ⊢ ¬ {x} = ∅
3		snprc 1838	. . . . . . . . . . 11 ⊢ (¬ A ∈ V ↔ {A} = ∅)
4		imaeq2 2603	. . . . . . . . . . 11 ⊢ ({A} = ∅ → (F “ {A}) = (F “ ∅))
5	3, 4	sylbi 174	. . . . . . . . . 10 ⊢ (¬ A ∈ V → (F “ {A}) = (F “ ∅))
6		ima0 2615	. . . . . . . . . 10 ⊢ (F “ ∅) = ∅
7	5, 6	syl6eq 1140	. . . . . . . . 9 ⊢ (¬ A ∈ V → (F “ {A}) = ∅)
8	7	cleq1d 1109	. . . . . . . 8 ⊢ (¬ A ∈ V → ((F “ {A}) = {x} ↔ ∅ = {x}))
9		cleqcom 1103	. . . . . . . 8 ⊢ (∅ = {x} ↔ {x} = ∅)
10	8, 9	syl6bb 414	. . . . . . 7 ⊢ (¬ A ∈ V → ((F “ {A}) = {x} ↔ {x} = ∅))
11	2, 10	mtbiri 539	. . . . . 6 ⊢ (¬ A ∈ V → ¬ (F “ {A}) = {x})
12	11	nexdv 983	. . . . 5 ⊢ (¬ A ∈ V → ¬ ∃x(F “ {A}) = {x})
13		abn0 1715	. . . . . 6 ⊢ (¬ {x∣(F “ {A}) = {x}} = ∅ ↔ ∃x(F “ {A}) = {x})
14	13	bicon1i 193	. . . . 5 ⊢ (¬ ∃x(F “ {A}) = {x} ↔ {x∣(F “ {A}) = {x}} = ∅)
15	12, 14	sylib 173	. . . 4 ⊢ (¬ A ∈ V → {x∣(F “ {A}) = {x}} = ∅)
16	15	unieqd 1929	. . 3 ⊢ (¬ A ∈ V → ∪{x∣(F “ {A}) = {x}} = ∪∅)
17		df-fv 2438	. . 3 ⊢ (F ‘A) = ∪{x∣(F “ {A}) = {x}}
18	16, 17	syl5eq 1136	. 2 ⊢ (¬ A ∈ V → (F ‘A) = ∪∅)
19		uni0 1938	. 2 ⊢ ∪∅ = ∅
20	18, 19	syl6eq 1140	1 ⊢ (¬ A ∈ V → (F ‘A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ∪cuni 1919   “ cima 2413   ‘cfv 2422

This theorem is referenced by:  tz6.12-2 2845  ndmfv 2848  fvopabn 2873  1stval 3089  2ndval 3090  rankon 3515  r1pwcl 3530  cardval 3633  sdomsdomcard 3654  cardcard 3655

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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