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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 11647 | . . . 4 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 11174 | . . 3 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1335 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | difss 4108 | . . . 4 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
7 | 6, 1 | eqeltri 2909 | . . . . 5 ⊢ 𝐴 ∈ ℕ |
8 | funsng 6405 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {〈𝐴, 𝑋〉}) | |
9 | 7, 8 | mpan 688 | . . . 4 ⊢ (𝑋 ∈ V → Fun {〈𝐴, 𝑋〉}) |
10 | funss 6374 | . . . 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 6419 | . . . 4 ⊢ Fun ∅ | |
13 | opprc2 4828 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → 〈𝐴, 𝑋〉 = ∅) | |
14 | 13 | sneqd 4579 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {〈𝐴, 𝑋〉} = {∅}) |
15 | 14 | difeq1d 4098 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ({∅} ∖ {∅})) |
16 | difid 4330 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
17 | 15, 16 | syl6eq 2872 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ∅) |
18 | 17 | funeqd 6377 | . . . 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 6071 | . . 3 ⊢ dom {〈𝐴, 𝑋〉} ⊆ {𝐴} | |
22 | 6 | sneqi 4578 | . . . 4 ⊢ {𝐴} = {𝐼} |
23 | 1 | nnzi 12007 | . . . . 5 ⊢ 𝐼 ∈ ℤ |
24 | fzsn 12950 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} |
26 | 22, 25 | eqtr4i 2847 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) |
27 | 21, 26 | sseqtri 4003 | . 2 ⊢ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼) |
28 | isstruct 16496 | . 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 3494 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 {csn 4567 〈cop 4573 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 (class class class)co 7156 ≤ cle 10676 ℕcn 11638 ℤcz 11982 ...cfz 12893 Struct cstr 16479 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 |
This theorem is referenced by: strle2 16593 strle3 16594 1strstr 16598 srngstr 16627 lmodstr 16636 phlstr 16653 cnfldstr 20547 |
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