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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 17222 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝑊‘(𝐸‘ndx))) |
| 4 | ovex 7464 | . . . . 5 ⊢ (𝑊 sSet 〈𝐷, 𝐶〉) ∈ V | |
| 5 | 4, 1 | strfvn 17223 | . . . 4 ⊢ (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) |
| 6 | setsres 17215 | . . . . . 6 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷})) = (𝑊 ↾ (V ∖ {𝐷}))) | |
| 7 | 6 | fveq1d 6908 | . . . . 5 ⊢ (𝑊 ∈ V → (((𝑊 sSet 〈𝐷, 𝐶〉) ↾ (V ∖ {𝐷}))‘(𝐸‘ndx)) = ((𝑊 ↾ (V ∖ {𝐷}))‘(𝐸‘ndx))) |
| 8 | fvex 6919 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ V | |
| 9 | setsnid.n | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ 𝐷 | |
| 10 | eldifsn 4786 | . . . . . . 7 ⊢ ((𝐸‘ndx) ∈ (V ∖ {𝐷}) ↔ ((𝐸‘ndx) ∈ V ∧ (𝐸‘ndx) ≠ 𝐷)) | |
| 11 | 8, 9, 10 | mpbir2an 711 | . . . . . 6 ⊢ (𝐸‘ndx) ∈ (V ∖ {𝐷}) |
| 12 | fvres 6925 | . . . . . 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 6925 | . . . . . 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 2800 | . . . 4 ⊢ (𝑊 ∈ V → ((𝑊 sSet 〈𝐷, 𝐶〉)‘(𝐸‘ndx)) = (𝑊‘(𝐸‘ndx))) |
| 17 | 5, 16 | eqtrid 2789 | . . 3 ⊢ (𝑊 ∈ V → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = (𝑊‘(𝐸‘ndx))) |
| 18 | 3, 17 | eqtr4d 2780 | . 2 ⊢ (𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
| 19 | 1 | str0 17226 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
| 20 | 19 | eqcomi 2746 | . . 3 ⊢ (𝐸‘∅) = ∅ |
| 21 | eqid 2737 | . . 3 ⊢ (𝑊 sSet 〈𝐷, 𝐶〉) = (𝑊 sSet 〈𝐷, 𝐶〉) | |
| 22 | reldmsets 17202 | . . 3 ⊢ Rel dom sSet | |
| 23 | 20, 21, 22 | oveqprc 17229 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 {csn 4626 〈cop 4632 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-slot 17219 |
| This theorem is referenced by: resseqnbas 17287 resslemOLD 17288 oppchomfval 17757 oppcbas 17761 rescbas 17873 rescco 17876 rescabs 17877 rescabsOLD 17878 odubas 18336 odubasOLD 18337 setsplusg 19368 oppglemOLD 19369 opprlem 20339 opprlemOLD 20340 rmodislmod 20928 sralem 21175 sralemOLD 21176 srasca 21183 srascaOLD 21184 sravsca 21185 sravscaOLD 21186 zlmlem 21527 zlmlemOLD 21528 zlmsca 21535 znbaslem 21553 znbaslemOLD 21554 thlbas 21714 thlbasOLD 21715 thlle 21716 thlleOLD 21717 opsrbaslem 22067 opsrbaslemOLD 22068 matbas 22417 matplusg 22418 matsca 22419 matscaOLD 22420 matvsca 22421 matvscaOLD 22422 tuslem 24275 tuslemOLD 24276 setsmsbas 24485 setsmsbasOLD 24486 setsmsds 24487 setsmsdsOLD 24488 tnglem 24653 tnglemOLD 24654 tngds 24668 tngdsOLD 24669 ttgval 28883 ttgvalOLD 28884 ttglem 28885 ttglemOLD 28886 cchhllem 28901 cchhllemOLD 28902 setsvtx 29052 resvlem 33357 resvlemOLD 33358 zlmds 33961 zlmdsOLD 33962 zlmtset 33963 zlmtsetOLD 33964 hlhilslem 41940 hlhilslemOLD 41941 mnringnmulrd 44228 mnringnmulrdOLD 44229 cznrnglem 48175 cznabel 48176 cznrng 48177 prstcnidlem 49154 prstcnid 49155 |
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