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Mirrors > Home > MPE Home > Th. List > Mathboxes > idsset | Structured version Visualization version GIF version |
Description: I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
idsset | ⊢ I = ( SSet ∩ ◡ SSet ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5830 | . 2 ⊢ Rel I | |
2 | relsset 35489 | . . 3 ⊢ Rel SSet | |
3 | relin1 5816 | . . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet )) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ Rel ( SSet ∩ ◡ SSet ) |
5 | eqss 3995 | . . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) | |
6 | vex 3475 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | 6 | ideq 5857 | . . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧) |
8 | brin 5202 | . . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧)) | |
9 | 6 | brsset 35490 | . . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧) |
10 | vex 3475 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | 10, 6 | brcnv 5887 | . . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦) |
12 | 10 | brsset 35490 | . . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦) |
13 | 11, 12 | bitri 274 | . . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦) |
14 | 9, 13 | anbi12i 626 | . . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) |
15 | 8, 14 | bitri 274 | . . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) |
16 | 5, 7, 15 | 3bitr4i 302 | . 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧) |
17 | 1, 4, 16 | eqbrriv 5795 | 1 ⊢ I = ( SSet ∩ ◡ SSet ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5150 I cid 5577 ◡ccnv 5679 Rel wrel 5685 SSet csset 35433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-eprel 5584 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fo 6557 df-fv 6559 df-1st 7997 df-2nd 7998 df-txp 35455 df-sset 35457 |
This theorem is referenced by: (None) |
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