Theorem fvopabn 2873

Description: This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class C it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvopabg 2872.

Hypothesis

Ref	Expression
fvopabg.1	⊢ (x = A → B = C)

Assertion

Ref	Expression
fvopabn	⊢ (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)

Distinct variable group(s):   x,y,A   y,B   x,C,y

Proof of Theorem fvopabn

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . 10 ⊢ z ∈ V
2	1	snnz 1846	. . . . . . . . 9 ⊢ ¬ {z} = ∅
3		opeq1 1876	. . . . . . . . . . . . . . . . . 18 ⊢ (z = A → ⟨z, w⟩ = ⟨A, w⟩)
4	3	eleq1d 1155	. . . . . . . . . . . . . . . . 17 ⊢ (z = A → (⟨z, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
5	4	ceqsexgv 1412	. . . . . . . . . . . . . . . 16 ⊢ (A ∈ V → (∃z(z = A ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
6		elsn 1820	. . . . . . . . . . . . . . . . . 18 ⊢ (z ∈ {A} ↔ z = A)
7	6	anbi1i 368	. . . . . . . . . . . . . . . . 17 ⊢ ((z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ (z = A ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}))
8	7	biex 733	. . . . . . . . . . . . . . . 16 ⊢ (∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ∃z(z = A ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}))
9	5, 8	syl5bb 410	. . . . . . . . . . . . . . 15 ⊢ (A ∈ V → (∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
10		visset 1350	. . . . . . . . . . . . . . . 16 ⊢ w ∈ V
11		fvopabg.1	. . . . . . . . . . . . . . . . . 18 ⊢ (x = A → B = C)
12	11	cleq2d 1112	. . . . . . . . . . . . . . . . 17 ⊢ (x = A → (y = B ↔ y = C))
13		cleq1 1107	. . . . . . . . . . . . . . . . 17 ⊢ (y = w → (y = C ↔ w = C))
14	12, 13	opelopabg 2115	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ V ∧ w ∈ V) → (⟨A, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ w = C))
15	10, 14	mpan2 519	. . . . . . . . . . . . . . 15 ⊢ (A ∈ V → (⟨A, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ w = C))
16	9, 15	bitrd 406	. . . . . . . . . . . . . 14 ⊢ (A ∈ V → (∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ w = C))
17	16	biabdv 1183	. . . . . . . . . . . . 13 ⊢ (A ∈ V → {w∣∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})} = {w∣w = C})
18		eleq1 1149	. . . . . . . . . . . . . . . . 17 ⊢ (w = C → (w ∈ V ↔ C ∈ V))
19	10, 18	mpbii 168	. . . . . . . . . . . . . . . 16 ⊢ (w = C → C ∈ V)
20	19	19.23aiv 952	. . . . . . . . . . . . . . 15 ⊢ (∃w w = C → C ∈ V)
21	20	con3i 90	. . . . . . . . . . . . . 14 ⊢ (¬ C ∈ V → ¬ ∃w w = C)
22		abn0 1715	. . . . . . . . . . . . . . 15 ⊢ (¬ {w∣w = C} = ∅ ↔ ∃w w = C)
23	22	bicon1i 193	. . . . . . . . . . . . . 14 ⊢ (¬ ∃w w = C ↔ {w∣w = C} = ∅)
24	21, 23	sylib 173	. . . . . . . . . . . . 13 ⊢ (¬ C ∈ V → {w∣w = C} = ∅)
25	17, 24	sylan9eq 1144	. . . . . . . . . . . 12 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → {w∣∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})} = ∅)
26		dfima3 2605	. . . . . . . . . . . 12 ⊢ ({⟨x, y⟩∣y = B} “ {A}) = {w∣∃z(z ∈ {A} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})}
27	25, 26	syl5eq 1136	. . . . . . . . . . 11 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} “ {A}) = ∅)
28	27	cleq1d 1109	. . . . . . . . . 10 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → (({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ ∅ = {z}))
29		cleqcom 1103	. . . . . . . . . 10 ⊢ (∅ = {z} ↔ {z} = ∅)
30	28, 29	syl6bb 414	. . . . . . . . 9 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → (({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ {z} = ∅))
31	2, 30	mtbiri 539	. . . . . . . 8 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ¬ ({⟨x, y⟩∣y = B} “ {A}) = {z})
32	31	19.21aiv 943	. . . . . . 7 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ∀z ¬ ({⟨x, y⟩∣y = B} “ {A}) = {z})
33		alnex 716	. . . . . . . 8 ⊢ (∀z ¬ ({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ ¬ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z})
34		abn0 1715	. . . . . . . . 9 ⊢ (¬ {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅ ↔ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z})
35	34	bicon1i 193	. . . . . . . 8 ⊢ (¬ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅)
36	33, 35	bitr 151	. . . . . . 7 ⊢ (∀z ¬ ({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅)
37	32, 36	sylib 173	. . . . . 6 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅)
38	37	unieqd 1929	. . . . 5 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ∪{z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∪∅)
39		df-fv 2438	. . . . 5 ⊢ ({⟨x, y⟩∣y = B} ‘A) = ∪{z∣({⟨x, y⟩∣y = B} “ {A}) = {z}}
40	38, 39	syl5eq 1136	. . . 4 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} ‘A) = ∪∅)
41		uni0 1938	. . . 4 ⊢ ∪∅ = ∅
42	40, 41	syl6eq 1140	. . 3 ⊢ ((A ∈ V ∧ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} ‘A) = ∅)
43	42	exp 291	. 2 ⊢ (A ∈ V → (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅))
44		fvprc 2829	. . 3 ⊢ (¬ A ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)
45	44	a1d 14	. 2 ⊢ (¬ A ∈ V → (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅))
46	43, 45	pm2.61i 110	1 ⊢ (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ⟨cop 1810  ∪cuni 1919  {copab 2055   “ cima 2413   ‘cfv 2422

This theorem is referenced by:  fvopabnf 2875

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-rex 1206  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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