Theorem fvopabn 3777

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 3776.

Hypothesis

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

Assertion

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

Distinct variable groups:   x,y,A   x,C,y

Proof of Theorem fvopabn

Step	Hyp	Ref	Expression
1		visset 1809	. . . . . . . . . . 11 ⊢ z ∈ V
2	1	snnz 2454	. . . . . . . . . 10 ⊢ {z} ≠ ∅
3		df-ne 1584	. . . . . . . . . 10 ⊢ ({z} ≠ ∅ ↔ ¬ {z} = ∅)
4	2, 3	mpbi 189	. . . . . . . . 9 ⊢ ¬ {z} = ∅
5		opeq1 2483	. . . . . . . . . . . . . . . . . 18 ⊢ (z = A → ⟨z, w⟩ = ⟨A, w⟩)
6	5	eleq1d 1537	. . . . . . . . . . . . . . . . 17 ⊢ (z = A → (⟨z, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
7	6	ceqsexgv 1884	. . . . . . . . . . . . . . . 16 ⊢ (A ∈ V → (∃z(z = A ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
8		elsn 2417	. . . . . . . . . . . . . . . . . 18 ⊢ (z ∈ {A} ↔ z = A)
9	8	anbi1i 481	. . . . . . . . . . . . . . . . 17 ⊢ ((z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ (z = A ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}))
10	9	exbii 1049	. . . . . . . . . . . . . . . 16 ⊢ (∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ∃z(z = A ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}))
11	7, 10	syl5bb 531	. . . . . . . . . . . . . . 15 ⊢ (A ∈ V → (∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ ⟨A, w⟩ ∈ {⟨x, y⟩∣y = B}))
12		visset 1809	. . . . . . . . . . . . . . . 16 ⊢ w ∈ V
13		fvopabn.1	. . . . . . . . . . . . . . . . . 18 ⊢ (x = A → B = C)
14	13	eqeq2d 1483	. . . . . . . . . . . . . . . . 17 ⊢ (x = A → (y = B ↔ y = C))
15		eqeq1 1478	. . . . . . . . . . . . . . . . 17 ⊢ (y = w → (y = C ↔ w = C))
16	14, 15	opelopabg 2812	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ V ⋀ w ∈ V) → (⟨A, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ w = C))
17	12, 16	mpan2 695	. . . . . . . . . . . . . . 15 ⊢ (A ∈ V → (⟨A, w⟩ ∈ {⟨x, y⟩∣y = B} ↔ w = C))
18	11, 17	bitrd 527	. . . . . . . . . . . . . 14 ⊢ (A ∈ V → (∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B}) ↔ w = C))
19	18	abbidv 1574	. . . . . . . . . . . . 13 ⊢ (A ∈ V → {w∣∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})} = {w∣w = C})
20		eleq1 1531	. . . . . . . . . . . . . . . . 17 ⊢ (w = C → (w ∈ V ↔ C ∈ V))
21	12, 20	mpbii 193	. . . . . . . . . . . . . . . 16 ⊢ (w = C → C ∈ V)
22	21	19.23aiv 1293	. . . . . . . . . . . . . . 15 ⊢ (∃w w = C → C ∈ V)
23	22	con3i 98	. . . . . . . . . . . . . 14 ⊢ (¬ C ∈ V → ¬ ∃w w = C)
24		abn0 2286	. . . . . . . . . . . . . . 15 ⊢ ({w∣w = C} ≠ ∅ ↔ ∃w w = C)
25	24	necon1bbii 1614	. . . . . . . . . . . . . 14 ⊢ (¬ ∃w w = C ↔ {w∣w = C} = ∅)
26	23, 25	sylib 198	. . . . . . . . . . . . 13 ⊢ (¬ C ∈ V → {w∣w = C} = ∅)
27	19, 26	sylan9eq 1524	. . . . . . . . . . . 12 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → {w∣∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})} = ∅)
28		dfima3 3398	. . . . . . . . . . . 12 ⊢ ({⟨x, y⟩∣y = B} “ {A}) = {w∣∃z(z ∈ {A} ⋀ ⟨z, w⟩ ∈ {⟨x, y⟩∣y = B})}
29	27, 28	syl5eq 1516	. . . . . . . . . . 11 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} “ {A}) = ∅)
30	29	eqeq1d 1480	. . . . . . . . . 10 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → (({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ ∅ = {z}))
31		eqcom 1474	. . . . . . . . . 10 ⊢ (∅ = {z} ↔ {z} = ∅)
32	30, 31	syl6bb 535	. . . . . . . . 9 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → (({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ {z} = ∅))
33	4, 32	mtbiri 716	. . . . . . . 8 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ¬ ({⟨x, y⟩∣y = B} “ {A}) = {z})
34	33	nexdv 1324	. . . . . . 7 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ¬ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z})
35		abn0 2286	. . . . . . . 8 ⊢ ({z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} ≠ ∅ ↔ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z})
36	35	necon1bbii 1614	. . . . . . 7 ⊢ (¬ ∃z({⟨x, y⟩∣y = B} “ {A}) = {z} ↔ {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅)
37	34, 36	sylib 198	. . . . . 6 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → {z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∅)
38	37	unieqd 2507	. . . . 5 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ∪{z∣({⟨x, y⟩∣y = B} “ {A}) = {z}} = ∪∅)
39		df-fv 3193	. . . . 5 ⊢ ({⟨x, y⟩∣y = B} ‘A) = ∪{z∣({⟨x, y⟩∣y = B} “ {A}) = {z}}
40	38, 39	syl5eq 1516	. . . 4 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} ‘A) = ∪∅)
41		uni0 2520	. . . 4 ⊢ ∪∅ = ∅
42	40, 41	syl6eq 1520	. . 3 ⊢ ((A ∈ V ⋀ ¬ C ∈ V) → ({⟨x, y⟩∣y = B} ‘A) = ∅)
43	42	ex 373	. 2 ⊢ (A ∈ V → (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅))
44		fvprc 3712	. . 3 ⊢ (¬ A ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)
45	44	a1d 12	. 2 ⊢ (¬ A ∈ V → (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅))
46	43, 45	pm2.61i 126	1 ⊢ (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978  {cab 1461   ≠ wne 1582  Vcvv 1807  ∅c0 2276  {csn 2405  ⟨cop 2407  ∪cuni 2498  {copab 2661   “ cima 3168   ‘cfv 3177

This theorem is referenced by:  fvopabnf 3779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

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