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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 11980 | . . . 4 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 11507 | . . 3 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1338 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | difss 4067 | . . . 4 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
7 | 6, 1 | eqeltri 2835 | . . . . 5 ⊢ 𝐴 ∈ ℕ |
8 | funsng 6487 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {〈𝐴, 𝑋〉}) | |
9 | 7, 8 | mpan 687 | . . . 4 ⊢ (𝑋 ∈ V → Fun {〈𝐴, 𝑋〉}) |
10 | funss 6455 | . . . 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 6501 | . . . 4 ⊢ Fun ∅ | |
13 | opprc2 4831 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → 〈𝐴, 𝑋〉 = ∅) | |
14 | 13 | sneqd 4575 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {〈𝐴, 𝑋〉} = {∅}) |
15 | 14 | difeq1d 4057 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ({∅} ∖ {∅})) |
16 | difid 4306 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
17 | 15, 16 | eqtrdi 2794 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ∅) |
18 | 17 | funeqd 6458 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ↔ Fun ∅)) |
19 | 12, 18 | mpbiri 257 | . . 3 ⊢ (¬ 𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) |
20 | 11, 19 | pm2.61i 182 | . 2 ⊢ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) |
21 | dmsnopss 6119 | . . 3 ⊢ dom {〈𝐴, 𝑋〉} ⊆ {𝐴} | |
22 | 6 | sneqi 4574 | . . . 4 ⊢ {𝐴} = {𝐼} |
23 | 1 | nnzi 12342 | . . . . 5 ⊢ 𝐼 ∈ ℤ |
24 | fzsn 13296 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} |
26 | 22, 25 | eqtr4i 2769 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) |
27 | 21, 26 | sseqtri 3958 | . 2 ⊢ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼) |
28 | isstruct 16851 | . 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 1340 | 1 ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3431 ∖ cdif 3885 ⊆ wss 3888 ∅c0 4258 {csn 4563 〈cop 4569 class class class wbr 5076 dom cdm 5591 Fun wfun 6429 (class class class)co 7277 ≤ cle 11008 ℕcn 11971 ℤcz 12317 ...cfz 13237 Struct cstr 16845 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 df-struct 16846 |
This theorem is referenced by: strle2 16858 strle3 16859 1strstr 16925 1strstr1 16926 srngstr 17017 lmodstr 17033 phlstr 17054 cnfldstr 20597 |
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