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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 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 ) |
Colors of variables: wff setvar class |
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) |
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