Theorem abrexexlem2 2911

Description: Lemma for abrexex 2912. Almost there, but still requires that B be a set.

Hypotheses

Ref	Expression
abrexexlem2.1	⊢ A ∈ V
abrexexlem2.2	⊢ B ∈ V

Assertion

Ref	Expression
abrexexlem2	⊢ {y∣∃x ∈ A y = B} ∈ V

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

Proof of Theorem abrexexlem2

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
2	1	biantrur 544	. . . . . . . . . 10 ⊢ (y = B ↔ (x ∈ V ∧ y = B))
3	2	biopabi 2103	. . . . . . . . 9 ⊢ {⟨x, y⟩∣y = B} = {⟨x, y⟩∣(x ∈ V ∧ y = B)}
4	3	fveq1i 2833	. . . . . . . 8 ⊢ ({⟨x, y⟩∣y = B} ‘x) = ({⟨x, y⟩∣(x ∈ V ∧ y = B)} ‘x)
5		abrexexlem2.2	. . . . . . . . 9 ⊢ B ∈ V
6		fvopab2 2878	. . . . . . . . 9 ⊢ ((x ∈ V ∧ B ∈ V) → ({⟨x, y⟩∣(x ∈ V ∧ y = B)} ‘x) = B)
7	1, 5, 6	mp2an 520	. . . . . . . 8 ⊢ ({⟨x, y⟩∣(x ∈ V ∧ y = B)} ‘x) = B
8	4, 7	eqtr 1119	. . . . . . 7 ⊢ ({⟨x, y⟩∣y = B} ‘x) = B
9	8	cleq2i 1111	. . . . . 6 ⊢ (y = ({⟨x, y⟩∣y = B} ‘x) ↔ y = B)
10	9	birex 1224	. . . . 5 ⊢ (∃x ∈ A y = ({⟨x, y⟩∣y = B} ‘x) ↔ ∃x ∈ A y = B)
11		ax-17 925	. . . . . 6 ⊢ (y = ({⟨x, y⟩∣y = B} ‘x) → ∀z y = ({⟨x, y⟩∣y = B} ‘x))
12		ax-17 925	. . . . . . 7 ⊢ (w ∈ y → ∀x w ∈ y)
13		hbopab1 2112	. . . . . . . 8 ⊢ (w ∈ {⟨x, y⟩∣y = B} → ∀x w ∈ {⟨x, y⟩∣y = B})
14		ax-17 925	. . . . . . . 8 ⊢ (w ∈ z → ∀x w ∈ z)
15	13, 14	hbfv 2837	. . . . . . 7 ⊢ (w ∈ ({⟨x, y⟩∣y = B} ‘z) → ∀x w ∈ ({⟨x, y⟩∣y = B} ‘z))
16	12, 15	hbeq 1171	. . . . . 6 ⊢ (y = ({⟨x, y⟩∣y = B} ‘z) → ∀x y = ({⟨x, y⟩∣y = B} ‘z))
17		fveq2 2832	. . . . . . 7 ⊢ (x = z → ({⟨x, y⟩∣y = B} ‘x) = ({⟨x, y⟩∣y = B} ‘z))
18	17	cleq2d 1112	. . . . . 6 ⊢ (x = z → (y = ({⟨x, y⟩∣y = B} ‘x) ↔ y = ({⟨x, y⟩∣y = B} ‘z)))
19	11, 16, 18	cbvrex 1332	. . . . 5 ⊢ (∃x ∈ A y = ({⟨x, y⟩∣y = B} ‘x) ↔ ∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z))
20	10, 19	bitr3 153	. . . 4 ⊢ (∃x ∈ A y = B ↔ ∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z))
21	20	biabi 1181	. . 3 ⊢ {y∣∃x ∈ A y = B} = {y∣∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z)}
22		ax-17 925	. . . 4 ⊢ (∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z) → ∀w∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z))
23		ax-17 925	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
24		ax-17 925	. . . . . 6 ⊢ (x ∈ w → ∀y x ∈ w)
25		hbopab2 2113	. . . . . . 7 ⊢ (w ∈ {⟨x, y⟩∣y = B} → ∀y w ∈ {⟨x, y⟩∣y = B})
26		ax-17 925	. . . . . . 7 ⊢ (w ∈ z → ∀y w ∈ z)
27	25, 26	hbfv 2837	. . . . . 6 ⊢ (w ∈ ({⟨x, y⟩∣y = B} ‘z) → ∀y w ∈ ({⟨x, y⟩∣y = B} ‘z))
28	24, 27	hbeq 1171	. . . . 5 ⊢ (w = ({⟨x, y⟩∣y = B} ‘z) → ∀y w = ({⟨x, y⟩∣y = B} ‘z))
29	23, 28	hbrex 1238	. . . 4 ⊢ (∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z) → ∀y∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z))
30		cleq1 1107	. . . . 5 ⊢ (y = w → (y = ({⟨x, y⟩∣y = B} ‘z) ↔ w = ({⟨x, y⟩∣y = B} ‘z)))
31	30	birexdv 1220	. . . 4 ⊢ (y = w → (∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z) ↔ ∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z)))
32	22, 29, 31	cbvab 1423	. . 3 ⊢ {y∣∃z ∈ A y = ({⟨x, y⟩∣y = B} ‘z)} = {w∣∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z)}
33	21, 32	eqtr 1119	. 2 ⊢ {y∣∃x ∈ A y = B} = {w∣∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z)}
34		abrexexlem2.1	. . 3 ⊢ A ∈ V
35	34	abrexexlem1 2910	. 2 ⊢ {w∣∃z ∈ A w = ({⟨x, y⟩∣y = B} ‘z)} ∈ V
36	33, 35	eqeltr 1159	1 ⊢ {y∣∃x ∈ A y = B} ∈ V

Syntax hints:   ∧ wa 196   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  {copab 2055   ‘cfv 2422

This theorem is referenced by:  abrexex 2912

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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