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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. (Contributed by AV, 14-Mar-2020.) |
Ref | Expression |
---|---|
setsidvald.e | ⊢ 𝐸 = Slot 𝑁 |
setsidvald.n | ⊢ 𝑁 ∈ ℕ |
setsidvald.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
setsidvald.f | ⊢ (𝜑 → Fun 𝑆) |
setsidvald.d | ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) |
Ref | Expression |
---|---|
setsidvald | ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsidvald.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
2 | fvex 6350 | . . 3 ⊢ (𝐸‘𝑆) ∈ V | |
3 | setsval 16061 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) | |
4 | 1, 2, 3 | sylancl 697 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
5 | setsidvald.e | . . . . . . 7 ⊢ 𝐸 = Slot 𝑁 | |
6 | setsidvald.n | . . . . . . 7 ⊢ 𝑁 ∈ ℕ | |
7 | 5, 6 | ndxid 16056 | . . . . . 6 ⊢ 𝐸 = Slot (𝐸‘ndx) |
8 | 7, 1 | strfvnd 16049 | . . . . 5 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
9 | 8 | opeq2d 4548 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
10 | 9 | sneqd 4321 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
11 | 10 | uneq2d 3898 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
12 | setsidvald.f | . . 3 ⊢ (𝜑 → Fun 𝑆) | |
13 | setsidvald.d | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) | |
14 | funresdfunsn 6607 | . . 3 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) | |
15 | 12, 13, 14 | syl2anc 696 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
16 | 4, 11, 15 | 3eqtrrd 2787 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1620 ∈ wcel 2127 Vcvv 3328 ∖ cdif 3700 ∪ cun 3701 {csn 4309 〈cop 4315 dom cdm 5254 ↾ cres 5256 Fun wfun 6031 ‘cfv 6037 (class class class)co 6801 ℕcn 11183 ndxcnx 16027 sSet csts 16028 Slot cslot 16029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-i2m1 10167 ax-1ne0 10168 ax-rrecex 10171 ax-cnre 10172 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-nn 11184 df-ndx 16033 df-slot 16034 df-sets 16037 |
This theorem is referenced by: ressval3d 16110 |
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