Theorem cnviinm 4782

Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
cnviinm	⊢ (∃y y ∈ A → ◡∩ x ∈ A B = ∩ x ∈ A ◡B)

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   B(x,y)

Proof of Theorem cnviinm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2078	. . 3 ⊢ (y = 𝑎 → (y ∈ A ↔ 𝑎 ∈ A))
2	1	cbvexv 1773	. 2 ⊢ (∃y y ∈ A ↔ ∃𝑎 𝑎 ∈ A)
3		eleq1 2078	. . . 4 ⊢ (x = 𝑎 → (x ∈ A ↔ 𝑎 ∈ A))
4	3	cbvexv 1773	. . 3 ⊢ (∃x x ∈ A ↔ ∃𝑎 𝑎 ∈ A)
5		relcnv 4626	. . . 4 ⊢ Rel ◡∩ x ∈ A B
6		r19.2m 3282	. . . . . . . 8 ⊢ ((∃x x ∈ A ∧ ∀x ∈ A ◡B ⊆ (V × V)) → ∃x ∈ A ◡B ⊆ (V × V))
7	6	expcom 109	. . . . . . 7 ⊢ (∀x ∈ A ◡B ⊆ (V × V) → (∃x x ∈ A → ∃x ∈ A ◡B ⊆ (V × V)))
8		relcnv 4626	. . . . . . . . 9 ⊢ Rel ◡B
9		df-rel 4275	. . . . . . . . 9 ⊢ (Rel ◡B ↔ ◡B ⊆ (V × V))
10	8, 9	mpbi 133	. . . . . . . 8 ⊢ ◡B ⊆ (V × V)
11	10	a1i 9	. . . . . . 7 ⊢ (x ∈ A → ◡B ⊆ (V × V))
12	7, 11	mprg 2352	. . . . . 6 ⊢ (∃x x ∈ A → ∃x ∈ A ◡B ⊆ (V × V))
13		iinss 3678	. . . . . 6 ⊢ (∃x ∈ A ◡B ⊆ (V × V) → ∩ x ∈ A ◡B ⊆ (V × V))
14	12, 13	syl 14	. . . . 5 ⊢ (∃x x ∈ A → ∩ x ∈ A ◡B ⊆ (V × V))
15		df-rel 4275	. . . . 5 ⊢ (Rel ∩ x ∈ A ◡B ↔ ∩ x ∈ A ◡B ⊆ (V × V))
16	14, 15	sylibr 137	. . . 4 ⊢ (∃x x ∈ A → Rel ∩ x ∈ A ◡B)
17		vex 2534	. . . . . . . 8 ⊢ 𝑏 ∈ V
18		vex 2534	. . . . . . . 8 ⊢ 𝑎 ∈ V
19	17, 18	opex 3936	. . . . . . 7 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
20		eliin 3632	. . . . . . 7 ⊢ (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ ∩ x ∈ A B ↔ ∀x ∈ A ⟨𝑏, 𝑎⟩ ∈ B))
21	19, 20	ax-mp 7	. . . . . 6 ⊢ (⟨𝑏, 𝑎⟩ ∈ ∩ x ∈ A B ↔ ∀x ∈ A ⟨𝑏, 𝑎⟩ ∈ B)
22	18, 17	opelcnv 4440	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ x ∈ A B ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ x ∈ A B)
23	18, 17	opex 3936	. . . . . . . 8 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
24		eliin 3632	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ ∩ x ∈ A ◡B ↔ ∀x ∈ A ⟨𝑎, 𝑏⟩ ∈ ◡B))
25	23, 24	ax-mp 7	. . . . . . 7 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ x ∈ A ◡B ↔ ∀x ∈ A ⟨𝑎, 𝑏⟩ ∈ ◡B)
26	18, 17	opelcnv 4440	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡B ↔ ⟨𝑏, 𝑎⟩ ∈ B)
27	26	ralbii 2304	. . . . . . 7 ⊢ (∀x ∈ A ⟨𝑎, 𝑏⟩ ∈ ◡B ↔ ∀x ∈ A ⟨𝑏, 𝑎⟩ ∈ B)
28	25, 27	bitri 173	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ x ∈ A ◡B ↔ ∀x ∈ A ⟨𝑏, 𝑎⟩ ∈ B)
29	21, 22, 28	3bitr4i 201	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ x ∈ A B ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ x ∈ A ◡B)
30	29	eqrelriv 4356	. . . 4 ⊢ ((Rel ◡∩ x ∈ A B ∧ Rel ∩ x ∈ A ◡B) → ◡∩ x ∈ A B = ∩ x ∈ A ◡B)
31	5, 16, 30	sylancr 395	. . 3 ⊢ (∃x x ∈ A → ◡∩ x ∈ A B = ∩ x ∈ A ◡B)
32	4, 31	sylbir 125	. 2 ⊢ (∃𝑎 𝑎 ∈ A → ◡∩ x ∈ A B = ∩ x ∈ A ◡B)
33	2, 32	sylbi 114	1 ⊢ (∃y y ∈ A → ◡∩ x ∈ A B = ∩ x ∈ A ◡B)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∀wral 2280  ∃wrex 2281  Vcvv 2531   ⊆ wss 2890  ⟨cop 3349  ∩ ciin 3628   × cxp 4266  ^◡ccnv 4267  Rel wrel 4273

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-iin 3630  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276

This theorem is referenced by: (None)