![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > strle1 | Structured version Visualization version GIF version |
Description: Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
Ref | Expression |
---|---|
strle1 | ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strle1.i | . . 3 ⊢ 𝐼 ∈ ℕ | |
2 | 1 | nnrei 11634 | . . . 4 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 11163 | . . 3 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1336 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | difss 4059 | . . . 4 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
7 | 6, 1 | eqeltri 2886 | . . . . 5 ⊢ 𝐴 ∈ ℕ |
8 | funsng 6375 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {〈𝐴, 𝑋〉}) | |
9 | 7, 8 | mpan 689 | . . . 4 ⊢ (𝑋 ∈ V → Fun {〈𝐴, 𝑋〉}) |
10 | funss 6343 | . . . 4 ⊢ (({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} → (Fun {〈𝐴, 𝑋〉} → Fun ({〈𝐴, 𝑋〉} ∖ {∅}))) | |
11 | 5, 9, 10 | mpsyl 68 | . . 3 ⊢ (𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) |
12 | fun0 6389 | . . . 4 ⊢ Fun ∅ | |
13 | opprc2 4790 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → 〈𝐴, 𝑋〉 = ∅) | |
14 | 13 | sneqd 4537 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {〈𝐴, 𝑋〉} = {∅}) |
15 | 14 | difeq1d 4049 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ({∅} ∖ {∅})) |
16 | difid 4284 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
17 | 15, 16 | eqtrdi 2849 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ∅) |
18 | 17 | funeqd 6346 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ↔ Fun ∅)) |
19 | 12, 18 | mpbiri 261 | . . 3 ⊢ (¬ 𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) |
20 | 11, 19 | pm2.61i 185 | . 2 ⊢ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) |
21 | dmsnopss 6038 | . . 3 ⊢ dom {〈𝐴, 𝑋〉} ⊆ {𝐴} | |
22 | 6 | sneqi 4536 | . . . 4 ⊢ {𝐴} = {𝐼} |
23 | 1 | nnzi 11994 | . . . . 5 ⊢ 𝐼 ∈ ℤ |
24 | fzsn 12944 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} |
26 | 22, 25 | eqtr4i 2824 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) |
27 | 21, 26 | sseqtri 3951 | . 2 ⊢ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼) |
28 | isstruct 16488 | . 2 ⊢ ({〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 ↔ ((𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) ∧ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ∧ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼))) | |
29 | 4, 20, 27, 28 | mpbir3an 1338 | 1 ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 {csn 4525 〈cop 4531 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 (class class class)co 7135 ≤ cle 10665 ℕcn 11625 ℤcz 11969 ...cfz 12885 Struct cstr 16471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 |
This theorem is referenced by: strle2 16585 strle3 16586 1strstr 16590 srngstr 16619 lmodstr 16628 phlstr 16645 cnfldstr 20093 |
Copyright terms: Public domain | W3C validator |