![]() |
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 12220 | . . . 4 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 11747 | . . 3 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1336 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | difss 4124 | . . . 4 ⊢ ({⟨𝐴, 𝑋⟩} ∖ {∅}) ⊆ {⟨𝐴, 𝑋⟩} | |
6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
7 | 6, 1 | eqeltri 2821 | . . . . 5 ⊢ 𝐴 ∈ ℕ |
8 | funsng 6590 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {⟨𝐴, 𝑋⟩}) | |
9 | 7, 8 | mpan 687 | . . . 4 ⊢ (𝑋 ∈ V → Fun {⟨𝐴, 𝑋⟩}) |
10 | funss 6558 | . . . 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 6604 | . . . 4 ⊢ Fun ∅ | |
13 | opprc2 4891 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → ⟨𝐴, 𝑋⟩ = ∅) | |
14 | 13 | sneqd 4633 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {⟨𝐴, 𝑋⟩} = {∅}) |
15 | 14 | difeq1d 4114 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({⟨𝐴, 𝑋⟩} ∖ {∅}) = ({∅} ∖ {∅})) |
16 | difid 4363 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
17 | 15, 16 | eqtrdi 2780 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({⟨𝐴, 𝑋⟩} ∖ {∅}) = ∅) |
18 | 17 | funeqd 6561 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (Fun ({⟨𝐴, 𝑋⟩} ∖ {∅}) ↔ Fun ∅)) |
19 | 12, 18 | mpbiri 258 | . . 3 ⊢ (¬ 𝑋 ∈ V → Fun ({⟨𝐴, 𝑋⟩} ∖ {∅})) |
20 | 11, 19 | pm2.61i 182 | . 2 ⊢ Fun ({⟨𝐴, 𝑋⟩} ∖ {∅}) |
21 | dmsnopss 6204 | . . 3 ⊢ dom {⟨𝐴, 𝑋⟩} ⊆ {𝐴} | |
22 | 6 | sneqi 4632 | . . . 4 ⊢ {𝐴} = {𝐼} |
23 | 1 | nnzi 12585 | . . . . 5 ⊢ 𝐼 ∈ ℤ |
24 | fzsn 13544 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} |
26 | 22, 25 | eqtr4i 2755 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) |
27 | 21, 26 | sseqtri 4011 | . 2 ⊢ dom {⟨𝐴, 𝑋⟩} ⊆ (𝐼...𝐼) |
28 | isstruct 17090 | . 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 1533 ∈ wcel 2098 Vcvv 3466 ∖ cdif 3938 ⊆ wss 3941 ∅c0 4315 {csn 4621 ⟨cop 4627 class class class wbr 5139 dom cdm 5667 Fun wfun 6528 (class class class)co 7402 ≤ cle 11248 ℕcn 12211 ℤcz 12557 ...cfz 13485 Struct cstr 17084 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 |
This theorem is referenced by: strle2 17097 strle3 17098 1strstr 17164 1strstr1 17165 srngstr 17259 lmodstr 17275 phlstr 17296 cnfldstr 21236 cnfldstrOLD 21251 |
Copyright terms: Public domain | W3C validator |