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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 8729 | . . . . 5 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 8247 | . . . 4 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1159 | . . 3 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | 4 | a1i 9 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼)) |
6 | difss 3202 | . . 3 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
7 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
8 | 7, 1 | eqeltri 2212 | . . . 4 ⊢ 𝐴 ∈ ℕ |
9 | funsng 5169 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → Fun {〈𝐴, 𝑋〉}) | |
10 | 8, 9 | mpan 420 | . . 3 ⊢ (𝑋 ∈ 𝑉 → Fun {〈𝐴, 𝑋〉}) |
11 | funss 5142 | . . 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 4150 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → 〈𝐴, 𝑋〉 ∈ V) | |
14 | 8, 13 | mpan 420 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 〈𝐴, 𝑋〉 ∈ V) |
15 | snexg 4108 | . . 3 ⊢ (〈𝐴, 𝑋〉 ∈ V → {〈𝐴, 𝑋〉} ∈ V) | |
16 | 14, 15 | syl 14 | . 2 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} ∈ V) |
17 | dmsnopg 5010 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = {𝐴}) | |
18 | 7 | sneqi 3539 | . . . . 5 ⊢ {𝐴} = {𝐼} |
19 | 1 | nnzi 9075 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
20 | fzsn 9846 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (𝐼...𝐼) = {𝐼} |
22 | 18, 21 | eqtr4i 2163 | . . . 4 ⊢ {𝐴} = (𝐼...𝐼) |
23 | 17, 22 | syl6eq 2188 | . . 3 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = (𝐼...𝐼)) |
24 | eqimss 3151 | . . 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 11974 | . 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 1218 | 1 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ⊆ wss 3071 ∅c0 3363 {csn 3527 〈cop 3530 class class class wbr 3929 dom cdm 4539 Fun wfun 5117 (class class class)co 5774 ≤ cle 7801 ℕcn 8720 ℤcz 9054 ...cfz 9790 Struct cstr 11955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-z 9055 df-uz 9327 df-fz 9791 df-struct 11961 |
This theorem is referenced by: strle2g 12050 strle3g 12051 1strstrg 12057 srngstrd 12081 lmodstrd 12092 |
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