Theorem fvopabnf 2875

Description: The value of a function given by an ordered pair abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvopabn 2873 uses bound variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
fvopabgf.1	⊢ (z ∈ A → ∀x z ∈ A)
fvopabgf.2	⊢ (z ∈ C → ∀x z ∈ C)
fvopabgf.3	⊢ (x = A → B = C)

Assertion

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

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

Proof of Theorem fvopabnf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (z ∈ w → ∀x z ∈ w)
2		fvopabgf.1	. . . . 5 ⊢ (z ∈ A → ∀x z ∈ A)
3		visset 1350	. . . . 5 ⊢ w ∈ V
4	1, 2, 3	eqvincf 1408	. . . 4 ⊢ (w = A ↔ ∃x(x = w ∧ x = A))
5		hbs1 986	. . . . . . 7 ⊢ ([w / x]u ∈ B → ∀x[w / x]u ∈ B)
6	5	hbab 1096	. . . . . 6 ⊢ (v ∈ {u∣[w / x]u ∈ B} → ∀x v ∈ {u∣[w / x]u ∈ B})
7		fvopabgf.2	. . . . . 6 ⊢ (z ∈ C → ∀x z ∈ C)
8	6, 7	hbeq 1171	. . . . 5 ⊢ ({u∣[w / x]u ∈ B} = C → ∀x{u∣[w / x]u ∈ B} = C)
9		sbab 1188	. . . . . . 7 ⊢ (x = w → B = {u∣[w / x]u ∈ B})
10	9	cleqcomd 1106	. . . . . 6 ⊢ (x = w → {u∣[w / x]u ∈ B} = B)
11		fvopabgf.3	. . . . . 6 ⊢ (x = A → B = C)
12	10, 11	sylan9eq 1144	. . . . 5 ⊢ ((x = w ∧ x = A) → {u∣[w / x]u ∈ B} = C)
13	8, 12	19.23ai 746	. . . 4 ⊢ (∃x(x = w ∧ x = A) → {u∣[w / x]u ∈ B} = C)
14	4, 13	sylbi 174	. . 3 ⊢ (w = A → {u∣[w / x]u ∈ B} = C)
15	14	fvopabn 2873	. 2 ⊢ (¬ C ∈ V → ({⟨w, v⟩∣v = {u∣[w / x]u ∈ B}} ‘A) = ∅)
16		ax-17 925	. . . 4 ⊢ (y = B → ∀w y = B)
17		ax-17 925	. . . 4 ⊢ (y = B → ∀v y = B)
18	6	hbeleq 1173	. . . 4 ⊢ (v = {u∣[w / x]u ∈ B} → ∀x v = {u∣[w / x]u ∈ B})
19		ax-17 925	. . . 4 ⊢ (v = {u∣[w / x]u ∈ B} → ∀y v = {u∣[w / x]u ∈ B})
20		id 9	. . . . 5 ⊢ (y = v → y = v)
21	20, 9	cleqan12rd 1117	. . . 4 ⊢ ((x = w ∧ y = v) → (y = B ↔ v = {u∣[w / x]u ∈ B}))
22	16, 17, 18, 19, 21	cbvopab 2104	. . 3 ⊢ {⟨x, y⟩∣y = B} = {⟨w, v⟩∣v = {u∣[w / x]u ∈ B}}
23	22	fveq1i 2833	. 2 ⊢ ({⟨x, y⟩∣y = B} ‘A) = ({⟨w, v⟩∣v = {u∣[w / x]u ∈ B}} ‘A)
24	15, 23	syl5eq 1136	1 ⊢ (¬ C ∈ V → ({⟨x, y⟩∣y = B} ‘A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {copab 2055   ‘cfv 2422

This theorem is referenced by:  rdgsucopabn 2985

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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