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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 8753 | . . . . 5 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 8271 | . . . 4 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1160 | . . 3 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | 4 | a1i 9 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼)) |
6 | difss 3207 | . . 3 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
7 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
8 | 7, 1 | eqeltri 2213 | . . . 4 ⊢ 𝐴 ∈ ℕ |
9 | funsng 5177 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → Fun {〈𝐴, 𝑋〉}) | |
10 | 8, 9 | mpan 421 | . . 3 ⊢ (𝑋 ∈ 𝑉 → Fun {〈𝐴, 𝑋〉}) |
11 | funss 5150 | . . 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 4158 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → 〈𝐴, 𝑋〉 ∈ V) | |
14 | 8, 13 | mpan 421 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 〈𝐴, 𝑋〉 ∈ V) |
15 | snexg 4116 | . . 3 ⊢ (〈𝐴, 𝑋〉 ∈ V → {〈𝐴, 𝑋〉} ∈ V) | |
16 | 14, 15 | syl 14 | . 2 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} ∈ V) |
17 | dmsnopg 5018 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = {𝐴}) | |
18 | 7 | sneqi 3544 | . . . . 5 ⊢ {𝐴} = {𝐼} |
19 | 1 | nnzi 9099 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
20 | fzsn 9877 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (𝐼...𝐼) = {𝐼} |
22 | 18, 21 | eqtr4i 2164 | . . . 4 ⊢ {𝐴} = (𝐼...𝐼) |
23 | 17, 22 | eqtrdi 2189 | . . 3 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = (𝐼...𝐼)) |
24 | eqimss 3156 | . . 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 12013 | . 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 1219 | 1 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∖ cdif 3073 ⊆ wss 3076 ∅c0 3368 {csn 3532 〈cop 3535 class class class wbr 3937 dom cdm 4547 Fun wfun 5125 (class class class)co 5782 ≤ cle 7825 ℕcn 8744 ℤcz 9078 ...cfz 9821 Struct cstr 11994 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-z 9079 df-uz 9351 df-fz 9822 df-struct 12000 |
This theorem is referenced by: strle2g 12089 strle3g 12090 1strstrg 12096 srngstrd 12120 lmodstrd 12131 |
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