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Mirrors > Home > ILE Home > Th. List > strle1g | GIF version |
Description: Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
Ref | Expression |
---|---|
strle1g | ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strle1.i | . . . 4 ⊢ 𝐼 ∈ ℕ | |
2 | 1 | nnrei 8931 | . . . . 5 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 8445 | . . . 4 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1175 | . . 3 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | 4 | a1i 9 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼)) |
6 | difss 3263 | . . 3 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
7 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
8 | 7, 1 | eqeltri 2250 | . . . 4 ⊢ 𝐴 ∈ ℕ |
9 | funsng 5264 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → Fun {〈𝐴, 𝑋〉}) | |
10 | 8, 9 | mpan 424 | . . 3 ⊢ (𝑋 ∈ 𝑉 → Fun {〈𝐴, 𝑋〉}) |
11 | funss 5237 | . . 3 ⊢ (({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} → (Fun {〈𝐴, 𝑋〉} → Fun ({〈𝐴, 𝑋〉} ∖ {∅}))) | |
12 | 6, 10, 11 | mpsyl 65 | . 2 ⊢ (𝑋 ∈ 𝑉 → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) |
13 | opexg 4230 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → 〈𝐴, 𝑋〉 ∈ V) | |
14 | 8, 13 | mpan 424 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 〈𝐴, 𝑋〉 ∈ V) |
15 | snexg 4186 | . . 3 ⊢ (〈𝐴, 𝑋〉 ∈ V → {〈𝐴, 𝑋〉} ∈ V) | |
16 | 14, 15 | syl 14 | . 2 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} ∈ V) |
17 | dmsnopg 5102 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = {𝐴}) | |
18 | 7 | sneqi 3606 | . . . . 5 ⊢ {𝐴} = {𝐼} |
19 | 1 | nnzi 9277 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
20 | fzsn 10069 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (𝐼...𝐼) = {𝐼} |
22 | 18, 21 | eqtr4i 2201 | . . . 4 ⊢ {𝐴} = (𝐼...𝐼) |
23 | 17, 22 | eqtrdi 2226 | . . 3 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = (𝐼...𝐼)) |
24 | eqimss 3211 | . . 3 ⊢ (dom {〈𝐴, 𝑋〉} = (𝐼...𝐼) → dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼)) | |
25 | 23, 24 | syl 14 | . 2 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼)) |
26 | isstructr 12480 | . 2 ⊢ (((𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) ∧ (Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ∧ {〈𝐴, 𝑋〉} ∈ V ∧ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼))) → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) | |
27 | 5, 12, 16, 25, 26 | syl13anc 1240 | 1 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 Vcvv 2739 ∖ cdif 3128 ⊆ wss 3131 ∅c0 3424 {csn 3594 〈cop 3597 class class class wbr 4005 dom cdm 4628 Fun wfun 5212 (class class class)co 5878 ≤ cle 7996 ℕcn 8922 ℤcz 9256 ...cfz 10011 Struct cstr 12461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-z 9257 df-uz 9532 df-fz 10012 df-struct 12467 |
This theorem is referenced by: strle2g 12569 strle3g 12570 1strstrg 12578 srngstrd 12607 lmodstrd 12625 |
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