Theorem cnviinm 5075

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		eleq1w 2198	. . 3 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
2	1	cbvexv 1890	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
3		eleq1w 2198	. . . 4 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
4	3	cbvexv 1890	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
5		relcnv 4912	. . . 4 ⊢ Rel ◡∩ 𝑥 ∈ 𝐴 𝐵
6		r19.2m 3444	. . . . . . . 8 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)) → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
7	6	expcom 115	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)))
8		relcnv 4912	. . . . . . . . 9 ⊢ Rel ◡𝐵
9		df-rel 4541	. . . . . . . . 9 ⊢ (Rel ◡𝐵 ↔ ◡𝐵 ⊆ (V × V))
10	8, 9	mpbi 144	. . . . . . . 8 ⊢ ◡𝐵 ⊆ (V × V)
11	10	a1i 9	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ◡𝐵 ⊆ (V × V))
12	7, 11	mprg 2487	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
13		iinss 3859	. . . . . 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 4541	. . . . 5 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
16	14, 15	sylibr 133	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵)
17		vex 2684	. . . . . . . 8 ⊢ 𝑏 ∈ V
18		vex 2684	. . . . . . . 8 ⊢ 𝑎 ∈ V
19	17, 18	opex 4146	. . . . . . 7 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
20		eliin 3813	. . . . . . 7 ⊢ (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
22	18, 17	opelcnv 4716	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
23	18, 17	opex 4146	. . . . . . . 8 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
24		eliin 3813	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵)
26	18, 17	opelcnv 4716	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
27	26	ralbii 2439	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
28	25, 27	bitri 183	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
29	21, 22, 28	3bitr4i 211	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵)
30	29	eqrelriv 4627	. . . 4 ⊢ ((Rel ◡∩ 𝑥 ∈ 𝐴 𝐵 ∧ Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵) → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
31	5, 16, 30	sylancr 410	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
32	4, 31	sylbir 134	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
33	2, 32	sylbi 120	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415  Vcvv 2681   ⊆ wss 3066  ⟨cop 3525  ∩ ciin 3809   × cxp 4532  ^◡ccnv 4533  Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iin 3811  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542

This theorem is referenced by: (None)