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Mirrors > Home > ILE Home > Th. List > setsslnid | GIF version |
Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
Ref | Expression |
---|---|
setsslid.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
setsslnid.n | ⊢ (𝐸‘ndx) ≠ 𝐷 |
setsslnid.d | ⊢ 𝐷 ∈ ℕ |
Ref | Expression |
---|---|
setsslnid | ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsslnid.d | . . . . 5 ⊢ 𝐷 ∈ ℕ | |
2 | setsresg 12656 | . . . . 5 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐷 ∈ ℕ ∧ 𝐶 ∈ 𝑉) → ((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷}))) | |
3 | 1, 2 | mp3an2 1336 | . . . 4 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → ((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷}))) |
4 | 3 | fveq1d 5556 | . . 3 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx))) |
5 | setsslid.e | . . . . . . 7 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
6 | 5 | simpri 113 | . . . . . 6 ⊢ (𝐸‘ndx) ∈ ℕ |
7 | 6 | elexi 2772 | . . . . 5 ⊢ (𝐸‘ndx) ∈ V |
8 | setsslnid.n | . . . . 5 ⊢ (𝐸‘ndx) ≠ 𝐷 | |
9 | eldifsn 3745 | . . . . 5 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) ↔ ((𝐸‘ndx) ∈ V ∧ (𝐸‘ndx) ≠ 𝐷)) | |
10 | 7, 8, 9 | mpbir2an 944 | . . . 4 ⊢ (𝐸‘ndx) ∈ (V ∖ {𝐷}) |
11 | fvres 5578 | . . . 4 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx))) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) |
13 | fvres 5578 | . . . 4 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) | |
14 | 10, 13 | ax-mp 5 | . . 3 ⊢ ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx)) |
15 | 4, 12, 14 | 3eqtr3g 2249 | . 2 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) |
16 | 5 | simpli 111 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
17 | setsex 12650 | . . . 4 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐷 ∈ ℕ ∧ 𝐶 ∈ 𝑉) → (𝑊 sSet 〈𝐷, 𝐶〉) ∈ V) | |
18 | 1, 17 | mp3an2 1336 | . . 3 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝑊 sSet 〈𝐷, 𝐶〉) ∈ V) |
19 | 6 | a1i 9 | . . 3 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘ndx) ∈ ℕ) |
20 | 16, 18, 19 | strnfvnd 12638 | . 2 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx))) |
21 | simpl 109 | . . 3 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝑊 ∈ 𝐴) | |
22 | 16, 21, 19 | strnfvnd 12638 | . 2 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘𝑊) = (𝑊‘(𝐸‘ndx))) |
23 | 15, 20, 22 | 3eqtr4rd 2237 | 1 ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 Vcvv 2760 ∖ cdif 3150 {csn 3618 〈cop 3621 ↾ cres 4661 ‘cfv 5254 (class class class)co 5918 ℕcn 8982 ndxcnx 12615 sSet csts 12616 Slot cslot 12617 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-iota 5215 df-fun 5256 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-slot 12622 df-sets 12625 |
This theorem is referenced by: resseqnbasd 12691 mgpbasg 13422 mgpscag 13423 mgptsetg 13424 mgpdsg 13426 opprsllem 13570 rmodislmod 13847 sralemg 13934 srascag 13938 sravscag 13939 zlmlemg 14116 zlmsca 14120 znbaslemnn 14127 setsmsbasg 14647 setsmsdsg 14648 |
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