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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 6904 | . . 3 ⊢ (𝐸‘𝑆) ∈ V | |
3 | setsval 17107 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝐸‘𝑆)〉})) | |
4 | 1, 2, 3 | sylancl 585 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝐸‘𝑆)〉})) |
5 | setsidvald.e | . . . . . 6 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5, 1 | strfvnd 17125 | . . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
7 | 6 | opeq2d 4880 | . . . 4 ⊢ (𝜑 → 〈𝑁, (𝐸‘𝑆)〉 = 〈𝑁, (𝑆‘𝑁)〉) |
8 | 7 | sneqd 4640 | . . 3 ⊢ (𝜑 → {〈𝑁, (𝐸‘𝑆)〉} = {〈𝑁, (𝑆‘𝑁)〉}) |
9 | 8 | uneq2d 4163 | . 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 7189 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝑁 ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝑆‘𝑁)〉}) = 𝑆) | |
13 | 10, 11, 12 | syl2anc 583 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {𝑁})) ∪ {〈𝑁, (𝑆‘𝑁)〉}) = 𝑆) |
14 | 4, 9, 13 | 3eqtrrd 2776 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 {csn 4628 〈cop 4634 dom cdm 5676 ↾ cres 5678 Fun wfun 6537 ‘cfv 6543 (class class class)co 7412 sSet csts 17103 Slot cslot 17121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-sets 17104 df-slot 17122 |
This theorem is referenced by: ressval3d 17198 opprabs 33038 |
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