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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.) (Proof shortened by AV, 7-Nov-2024.) |
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 17118 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝑊‘(𝐸‘ndx))) |
4 | ovex 7442 | . . . . 5 ⊢ (𝑊 sSet 〈𝐷, 𝐶〉) ∈ V | |
5 | 4, 1 | strfvn 17119 | . . . 4 ⊢ (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) |
6 | setsres 17111 | . . . . . 6 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷}))) | |
7 | 6 | fveq1d 6894 | . . . . 5 ⊢ (𝑊 ∈ V → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx))) |
8 | fvex 6905 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ V | |
9 | setsnid.n | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ 𝐷 | |
10 | eldifsn 4791 | . . . . . . 7 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) ↔ ((𝐸‘ndx) ∈ V ∧ (𝐸‘ndx) ≠ 𝐷)) | |
11 | 8, 9, 10 | mpbir2an 710 | . . . . . 6 ⊢ (𝐸‘ndx) ∈ (V ∖ {𝐷}) |
12 | fvres 6911 | . . . . . 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 6911 | . . . . . 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 2796 | . . . 4 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) |
17 | 5, 16 | eqtrid 2785 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = (𝑊‘(𝐸‘ndx))) |
18 | 3, 17 | eqtr4d 2776 | . 2 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
19 | 1 | str0 17122 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
20 | 19 | eqcomi 2742 | . . 3 ⊢ (𝐸‘∅) = ∅ |
21 | eqid 2733 | . . 3 ⊢ (𝑊 sSet 〈𝐷, 𝐶〉) = (𝑊 sSet 〈𝐷, 𝐶〉) | |
22 | reldmsets 17098 | . . 3 ⊢ Rel dom sSet | |
23 | 20, 21, 22 | oveqprc 17125 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
24 | 18, 23 | pm2.61i 182 | 1 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3946 ∅c0 4323 {csn 4629 〈cop 4635 ↾ cres 5679 ‘cfv 6544 (class class class)co 7409 sSet csts 17096 Slot cslot 17114 ndxcnx 17126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-res 5689 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-sets 17097 df-slot 17115 |
This theorem is referenced by: resseqnbas 17186 resslemOLD 17187 oppchomfval 17658 oppchomfvalOLD 17659 oppcbas 17663 oppcbasOLD 17664 rescbas 17776 rescbasOLD 17777 rescco 17780 resccoOLD 17781 rescabs 17782 rescabsOLD 17783 odubas 18244 odubasOLD 18245 setsplusg 19214 oppglemOLD 19215 mgplemOLD 19992 opprlem 20155 opprlemOLD 20156 rmodislmod 20540 rmodislmodOLD 20541 sralem 20790 sralemOLD 20791 srasca 20798 srascaOLD 20799 sravsca 20800 sravscaOLD 20801 zlmlem 21066 zlmlemOLD 21067 zlmsca 21074 znbaslem 21090 znbaslemOLD 21091 thlbas 21249 thlbasOLD 21250 thlle 21251 thlleOLD 21252 opsrbaslem 21604 opsrbaslemOLD 21605 matbas 21913 matplusg 21914 matsca 21915 matscaOLD 21916 matvsca 21917 matvscaOLD 21918 tuslem 23771 tuslemOLD 23772 setsmsbas 23981 setsmsbasOLD 23982 setsmsds 23983 setsmsdsOLD 23984 tnglem 24149 tnglemOLD 24150 tngds 24164 tngdsOLD 24165 ttgval 28126 ttgvalOLD 28127 ttglem 28128 ttglemOLD 28129 cchhllem 28144 cchhllemOLD 28145 setsvtx 28295 resvlem 32445 resvlemOLD 32446 zlmds 32942 zlmdsOLD 32943 zlmtset 32944 zlmtsetOLD 32945 hlhilslem 40809 hlhilslemOLD 40810 mnringnmulrd 42968 mnringnmulrdOLD 42969 cznrnglem 46851 cznabel 46852 cznrng 46853 prstcnidlem 47685 prstcnid 47686 |
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