Theorem idsset 33353

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 5700	. 2 ⊢ Rel I
2		relsset 33351	. . 3 ⊢ Rel SSet
3		relin1 5687	. . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet ))
4	2, 3	ax-mp 5	. 2 ⊢ Rel ( SSet ∩ ◡ SSet )
5		eqss 3984	. . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
6		vex 3499	. . . 4 ⊢ 𝑧 ∈ V
7	6	ideq 5725	. . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
8		brin 5120	. . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧))
9	6	brsset 33352	. . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10		vex 3499	. . . . . . 7 ⊢ 𝑦 ∈ V
11	10, 6	brcnv 5755	. . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦)
12	10	brsset 33352	. . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦)
13	11, 12	bitri 277	. . . . 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 277	. . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
16	5, 7, 15	3bitr4i 305	. 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧)
17	1, 4, 16	eqbrriv 5666	1 ⊢ I = ( SSet ∩ ◡ SSet )

Syntax hints:   ∧ wa 398   = wceq 1537   ∩ cin 3937   ⊆ wss 3938   class class class wbr 5068   I cid 5461  ^◡ccnv 5556  Rel wrel 5562   SSet csset 33295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-eprel 5467  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-1st 7691  df-2nd 7692  df-txp 33317  df-sset 33319

This theorem is referenced by: (None)