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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 5839 | . 2 ⊢ Rel I | |
2 | relsset 35870 | . . 3 ⊢ Rel SSet | |
3 | relin1 5825 | . . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet )) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ Rel ( SSet ∩ ◡ SSet ) |
5 | eqss 4011 | . . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) | |
6 | vex 3482 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | 6 | ideq 5866 | . . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧) |
8 | brin 5200 | . . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧)) | |
9 | 6 | brsset 35871 | . . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧) |
10 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | 10, 6 | brcnv 5896 | . . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦) |
12 | 10 | brsset 35871 | . . . . . 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 5804 | 1 ⊢ I = ( SSet ∩ ◡ SSet ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 I cid 5582 ◡ccnv 5688 Rel wrel 5694 SSet csset 35814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-txp 35836 df-sset 35838 |
This theorem is referenced by: (None) |
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