Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8858 | . . . . 5 ⊢ 𝐼 ∈ ℝ |
3 | 2 | leidi 8375 | . . . 4 ⊢ 𝐼 ≤ 𝐼 |
4 | 1, 1, 3 | 3pm3.2i 1164 | . . 3 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) |
5 | 4 | a1i 9 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼)) |
6 | difss 3244 | . . 3 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
7 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
8 | 7, 1 | eqeltri 2237 | . . . 4 ⊢ 𝐴 ∈ ℕ |
9 | funsng 5229 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → Fun {〈𝐴, 𝑋〉}) | |
10 | 8, 9 | mpan 421 | . . 3 ⊢ (𝑋 ∈ 𝑉 → Fun {〈𝐴, 𝑋〉}) |
11 | funss 5202 | . . 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 4201 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → 〈𝐴, 𝑋〉 ∈ V) | |
14 | 8, 13 | mpan 421 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 〈𝐴, 𝑋〉 ∈ V) |
15 | snexg 4158 | . . 3 ⊢ (〈𝐴, 𝑋〉 ∈ V → {〈𝐴, 𝑋〉} ∈ V) | |
16 | 14, 15 | syl 14 | . 2 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} ∈ V) |
17 | dmsnopg 5070 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = {𝐴}) | |
18 | 7 | sneqi 3583 | . . . . 5 ⊢ {𝐴} = {𝐼} |
19 | 1 | nnzi 9204 | . . . . . 6 ⊢ 𝐼 ∈ ℤ |
20 | fzsn 9992 | . . . . . 6 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (𝐼...𝐼) = {𝐼} |
22 | 18, 21 | eqtr4i 2188 | . . . 4 ⊢ {𝐴} = (𝐼...𝐼) |
23 | 17, 22 | eqtrdi 2213 | . . 3 ⊢ (𝑋 ∈ 𝑉 → dom {〈𝐴, 𝑋〉} = (𝐼...𝐼)) |
24 | eqimss 3192 | . . 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 12372 | . 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 1229 | 1 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 Vcvv 2722 ∖ cdif 3109 ⊆ wss 3112 ∅c0 3405 {csn 3571 〈cop 3574 class class class wbr 3977 dom cdm 4599 Fun wfun 5177 (class class class)co 5837 ≤ cle 7926 ℕcn 8849 ℤcz 9183 ...cfz 9936 Struct cstr 12353 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-z 9184 df-uz 9459 df-fz 9937 df-struct 12359 |
This theorem is referenced by: strle2g 12448 strle3g 12449 1strstrg 12455 srngstrd 12479 lmodstrd 12490 |
Copyright terms: Public domain | W3C validator |