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| Mirrors > Home > MPE Home > Th. List > setsidvald | Structured version Visualization version GIF version | ||
| Description: Value of the structure
replacement function, deduction version. Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| setsidvald.e | ⊢ 𝐸 = Slot 𝑁 | 
| setsidvald.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) | 
| setsidvald.f | ⊢ (𝜑 → Fun 𝑆) | 
| setsidvald.d | ⊢ (𝜑 → 𝑁 ∈ dom 𝑆) | 
| Ref | Expression | 
|---|---|
| setsidvald | ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | setsidvald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
| 2 | fvex 6919 | . . 3 ⊢ (𝐸‘𝑆) ∈ V | |
| 3 | setsval 17204 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝐸‘𝑆)〉})) | |
| 4 | 1, 2, 3 | sylancl 586 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝐸‘𝑆)〉})) | 
| 5 | setsidvald.e | . . . . . 6 ⊢ 𝐸 = Slot 𝑁 | |
| 6 | 5, 1 | strfvnd 17222 | . . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) | 
| 7 | 6 | opeq2d 4880 | . . . 4 ⊢ (𝜑 → 〈𝑁, (𝐸‘𝑆)〉 = 〈𝑁, (𝑆‘𝑁)〉) | 
| 8 | 7 | sneqd 4638 | . . 3 ⊢ (𝜑 → {〈𝑁, (𝐸‘𝑆)〉} = {〈𝑁, (𝑆‘𝑁)〉}) | 
| 9 | 8 | uneq2d 4168 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝑆‘𝑁)〉})) | 
| 10 | setsidvald.f | . . 3 ⊢ (𝜑 → Fun 𝑆) | |
| 11 | setsidvald.d | . . 3 ⊢ (𝜑 → 𝑁 ∈ dom 𝑆) | |
| 12 | funresdfunsn 7209 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝑆‘𝑁)〉}) = 𝑆) | |
| 13 | 10, 11, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝑆‘𝑁)〉}) = 𝑆) | 
| 14 | 4, 9, 13 | 3eqtrrd 2782 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 {csn 4626 〈cop 4632 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 Slot cslot 17218 | 
| 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-reu 3381 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-slot 17219 | 
| This theorem is referenced by: ressval3d 17292 opprabs 33510 | 
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