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Mirrors > Home > MPE Home > Th. List > structcnvcnv | Structured version Visualization version GIF version |
Description: Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
structcnvcnv | ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5671 | . . . . . 6 ⊢ ¬ ∅ ∈ (V × V) | |
2 | cnvcnv 6148 | . . . . . . . 8 ⊢ ◡◡𝐹 = (𝐹 ∩ (V × V)) | |
3 | inss2 4193 | . . . . . . . 8 ⊢ (𝐹 ∩ (V × V)) ⊆ (V × V) | |
4 | 2, 3 | eqsstri 3982 | . . . . . . 7 ⊢ ◡◡𝐹 ⊆ (V × V) |
5 | 4 | sseli 3944 | . . . . . 6 ⊢ (∅ ∈ ◡◡𝐹 → ∅ ∈ (V × V)) |
6 | 1, 5 | mto 196 | . . . . 5 ⊢ ¬ ∅ ∈ ◡◡𝐹 |
7 | disjsn 4676 | . . . . 5 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡◡𝐹) | |
8 | 6, 7 | mpbir 230 | . . . 4 ⊢ (◡◡𝐹 ∩ {∅}) = ∅ |
9 | cnvcnvss 6150 | . . . . 5 ⊢ ◡◡𝐹 ⊆ 𝐹 | |
10 | reldisj 4415 | . . . . 5 ⊢ (◡◡𝐹 ⊆ 𝐹 → ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}))) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ((◡◡𝐹 ∩ {∅}) = ∅ ↔ ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
12 | 8, 11 | mpbi 229 | . . 3 ⊢ ◡◡𝐹 ⊆ (𝐹 ∖ {∅}) |
13 | 12 | a1i 11 | . 2 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 ⊆ (𝐹 ∖ {∅})) |
14 | structn0fun 17031 | . . . . 5 ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | |
15 | funrel 6522 | . . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) → Rel (𝐹 ∖ {∅})) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝐹 Struct 𝑋 → Rel (𝐹 ∖ {∅})) |
17 | dfrel2 6145 | . . . 4 ⊢ (Rel (𝐹 ∖ {∅}) ↔ ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) | |
18 | 16, 17 | sylib 217 | . . 3 ⊢ (𝐹 Struct 𝑋 → ◡◡(𝐹 ∖ {∅}) = (𝐹 ∖ {∅})) |
19 | difss 4095 | . . . 4 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹 | |
20 | cnvss 5832 | . . . 4 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → ◡(𝐹 ∖ {∅}) ⊆ ◡𝐹) | |
21 | cnvss 5832 | . . . 4 ⊢ (◡(𝐹 ∖ {∅}) ⊆ ◡𝐹 → ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹) | |
22 | 19, 20, 21 | mp2b 10 | . . 3 ⊢ ◡◡(𝐹 ∖ {∅}) ⊆ ◡◡𝐹 |
23 | 18, 22 | eqsstrrdi 4003 | . 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∖ {∅}) ⊆ ◡◡𝐹) |
24 | 13, 23 | eqssd 3965 | 1 ⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 {csn 4590 class class class wbr 5109 × cxp 5635 ◡ccnv 5636 Rel wrel 5642 Fun wfun 6494 Struct cstr 17026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 |
This theorem is referenced by: structfung 17034 ebtwntg 27980 ecgrtg 27981 elntg 27982 |
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