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| Mirrors > Home > MPE Home > Th. List > setsnidOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of setsnid 17245 as of 7-Nov-2024. Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| setsid.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
| setsnid.n | ⊢ (𝐸‘ndx) ≠ 𝐷 |
| Ref | Expression |
|---|---|
| setsnidOLD | ⊢ (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 | . . 3 ⊢ ∅ = (𝐸‘∅) |
| 20 | fvprc 6898 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
| 21 | reldmsets 17202 | . . . . 5 ⊢ Rel dom sSet | |
| 22 | 21 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 sSet 〈𝐷, 𝐶〉) = ∅) |
| 23 | 22 | fveq2d 6910 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) = (𝐸‘∅)) |
| 24 | 19, 20, 23 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) |
| 25 | 18, 24 | pm2.61i 182 | 1 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = 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: (None) |
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