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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 6920 | . . 3 ⊢ (𝐸‘𝑆) ∈ V | |
3 | setsval 17201 | . . 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 17219 | . . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
7 | 6 | opeq2d 4885 | . . . 4 ⊢ (𝜑 → 〈𝑁, (𝐸‘𝑆)〉 = 〈𝑁, (𝑆‘𝑁)〉) |
8 | 7 | sneqd 4643 | . . 3 ⊢ (𝜑 → {〈𝑁, (𝐸‘𝑆)〉} = {〈𝑁, (𝑆‘𝑁)〉}) |
9 | 8 | uneq2d 4178 | . 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 2780 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈𝑁, (𝐸‘𝑆)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 Slot cslot 17215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-slot 17216 |
This theorem is referenced by: ressval3d 17292 opprabs 33490 |
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