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Mirrors > Home > MPE Home > Th. List > setsnid | Structured version Visualization version GIF version |
Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
setsid.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
setsnid.n | ⊢ (𝐸‘ndx) ≠ 𝐷 |
Ref | Expression |
---|---|
setsnid | ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsid.e | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | id 22 | . . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V) | |
3 | 1, 2 | strfvnd 16494 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝑊‘(𝐸‘ndx))) |
4 | ovex 7168 | . . . . 5 ⊢ (𝑊 sSet 〈𝐷, 𝐶〉) ∈ V | |
5 | 4, 1 | strfvn 16497 | . . . 4 ⊢ (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) |
6 | setsres 16517 | . . . . . 6 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷}))) | |
7 | 6 | fveq1d 6647 | . . . . 5 ⊢ (𝑊 ∈ V → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx))) |
8 | fvex 6658 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ V | |
9 | setsnid.n | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ 𝐷 | |
10 | eldifsn 4680 | . . . . . . 7 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) ↔ ((𝐸‘ndx) ∈ V ∧ (𝐸‘ndx) ≠ 𝐷)) | |
11 | 8, 9, 10 | mpbir2an 710 | . . . . . 6 ⊢ (𝐸‘ndx) ∈ (V ∖ {𝐷}) |
12 | fvres 6664 | . . . . . 6 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) |
14 | fvres 6664 | . . . . . 6 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) → ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) | |
15 | 11, 14 | ax-mp 5 | . . . . 5 ⊢ ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx)) |
16 | 7, 13, 15 | 3eqtr3g 2856 | . . . 4 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) |
17 | 5, 16 | syl5eq 2845 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = (𝑊‘(𝐸‘ndx))) |
18 | 3, 17 | eqtr4d 2836 | . 2 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
19 | 1 | str0 16527 | . . 3 ⊢ ∅ = (𝐸‘∅) |
20 | fvprc 6638 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
21 | reldmsets 16503 | . . . . 5 ⊢ Rel dom sSet | |
22 | 21 | ovprc1 7174 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 sSet 〈𝐷, 𝐶〉) = ∅) |
23 | 22 | fveq2d 6649 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = (𝐸‘∅)) |
24 | 19, 20, 23 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
25 | 18, 24 | pm2.61i 185 | 1 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 {csn 4525 〈cop 4531 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 Slot cslot 16474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-slot 16479 df-sets 16482 |
This theorem is referenced by: resslem 16549 oppchomfval 16976 oppcbas 16980 rescbas 17091 rescco 17094 rescabs 17095 odubas 17735 oppglem 18470 mgplem 19237 opprlem 19374 rmodislmod 19695 sralem 19942 srasca 19946 sravsca 19947 zlmlem 20210 zlmsca 20214 znbaslem 20230 thlbas 20385 thlle 20386 opsrbaslem 20717 matbas 21018 matplusg 21019 matsca 21020 matvsca 21021 tuslem 22873 setsmsbas 23082 setsmsds 23083 tnglem 23246 tngds 23254 ttgval 26669 ttglem 26670 cchhllem 26681 setsvtx 26828 resvlem 30955 zlmds 31315 zlmtset 31316 hlhilslem 39234 mnringnmulrd 40922 cznrnglem 44577 cznabel 44578 cznrng 44579 |
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