Theorem idsset 35878

Description: I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
idsset	⊢ I = ( SSet ∩ ◡ SSet )

Proof of Theorem idsset

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reli 5789	. 2 ⊢ Rel I
2		relsset 35876	. . 3 ⊢ Rel SSet
3		relin1 5775	. . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet ))
4	2, 3	ax-mp 5	. 2 ⊢ Rel ( SSet ∩ ◡ SSet )
5		eqss 3962	. . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
6		vex 3451	. . . 4 ⊢ 𝑧 ∈ V
7	6	ideq 5816	. . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
8		brin 5159	. . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧))
9	6	brsset 35877	. . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10		vex 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
11	10, 6	brcnv 5846	. . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦)
12	10	brsset 35877	. . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦)
13	11, 12	bitri 275	. . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦)
14	9, 13	anbi12i 628	. . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
15	8, 14	bitri 275	. . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
16	5, 7, 15	3bitr4i 303	. 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧)
17	1, 4, 16	eqbrriv 5754	1 ⊢ I = ( SSet ∩ ◡ SSet )

Syntax hints:   ∧ wa 395   = wceq 1540   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107   I cid 5532  ^◡ccnv 5637  Rel wrel 5643   SSet csset 35820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842  df-sset 35844

This theorem is referenced by: (None)