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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 11649 | . . . 4 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 11176 | . . 3 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1335 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | difss 4110 | . . . 4 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
7 | 6, 1 | eqeltri 2911 | . . . . 5 ⊢ 𝐴 ∈ ℕ |
8 | funsng 6407 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {〈𝐴, 𝑋〉}) | |
9 | 7, 8 | mpan 688 | . . . 4 ⊢ (𝑋 ∈ V → Fun {〈𝐴, 𝑋〉}) |
10 | funss 6376 | . . . 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 6421 | . . . 4 ⊢ Fun ∅ | |
13 | opprc2 4830 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → 〈𝐴, 𝑋〉 = ∅) | |
14 | 13 | sneqd 4581 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {〈𝐴, 𝑋〉} = {∅}) |
15 | 14 | difeq1d 4100 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ({∅} ∖ {∅})) |
16 | difid 4332 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
17 | 15, 16 | syl6eq 2874 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ∅) |
18 | 17 | funeqd 6379 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ↔ Fun ∅)) |
19 | 12, 18 | mpbiri 260 | . . 3 ⊢ (¬ 𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) |
20 | 11, 19 | pm2.61i 184 | . 2 ⊢ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) |
21 | dmsnopss 6073 | . . 3 ⊢ dom {〈𝐴, 𝑋〉} ⊆ {𝐴} | |
22 | 6 | sneqi 4580 | . . . 4 ⊢ {𝐴} = {𝐼} |
23 | 1 | nnzi 12009 | . . . . 5 ⊢ 𝐼 ∈ ℤ |
24 | fzsn 12952 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} |
26 | 22, 25 | eqtr4i 2849 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) |
27 | 21, 26 | sseqtri 4005 | . 2 ⊢ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼) |
28 | isstruct 16498 | . 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 1337 | 1 ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 class class class wbr 5068 dom cdm 5557 Fun wfun 6351 (class class class)co 7158 ≤ cle 10678 ℕcn 11640 ℤcz 11984 ...cfz 12895 Struct cstr 16481 |
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 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 |
This theorem is referenced by: strle2 16595 strle3 16596 1strstr 16600 srngstr 16629 lmodstr 16638 phlstr 16655 cnfldstr 20549 |
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