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Mirrors > Home > ILE Home > Th. List > strsetsid | GIF version |
Description: Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strsetsid.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
strsetsid.s | ⊢ (𝜑 → 𝑆 Struct 〈𝑀, 𝑁〉) |
strsetsid.f | ⊢ (𝜑 → Fun 𝑆) |
strsetsid.d | ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) |
Ref | Expression |
---|---|
strsetsid | ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strsetsid.s | . . . 4 ⊢ (𝜑 → 𝑆 Struct 〈𝑀, 𝑁〉) | |
2 | structex 12527 | . . . 4 ⊢ (𝑆 Struct 〈𝑀, 𝑁〉 → 𝑆 ∈ V) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
4 | strsetsid.d | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) | |
5 | strsetsid.e | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
6 | isstructim 12529 | . . . . . . . . 9 ⊢ (𝑆 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝑆 ∖ {∅}) ∧ dom 𝑆 ⊆ (𝑀...𝑁))) | |
7 | 1, 6 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝑆 ∖ {∅}) ∧ dom 𝑆 ⊆ (𝑀...𝑁))) |
8 | 7 | simp3d 1013 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 ⊆ (𝑀...𝑁)) |
9 | 7 | simp1d 1011 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) |
10 | 9 | simp1d 1011 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | fzssnn 10100 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | |
12 | 10, 11 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀...𝑁) ⊆ ℕ) |
13 | 8, 12 | sstrd 3180 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 ⊆ ℕ) |
14 | 13, 4 | sseldd 3171 | . . . . 5 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
15 | 5, 3, 14 | strnfvnd 12535 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
16 | strsetsid.f | . . . . 5 ⊢ (𝜑 → Fun 𝑆) | |
17 | funfvex 5551 | . . . . 5 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → (𝑆‘(𝐸‘ndx)) ∈ V) | |
18 | 16, 4, 17 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) ∈ V) |
19 | 15, 18 | eqeltrd 2266 | . . 3 ⊢ (𝜑 → (𝐸‘𝑆) ∈ V) |
20 | setsvala 12546 | . . 3 ⊢ ((𝑆 ∈ V ∧ (𝐸‘ndx) ∈ dom 𝑆 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) | |
21 | 3, 4, 19, 20 | syl3anc 1249 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
22 | 15 | opeq2d 3800 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
23 | 22 | sneqd 3620 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
24 | 23 | uneq2d 3304 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
25 | nnssz 9301 | . . . . 5 ⊢ ℕ ⊆ ℤ | |
26 | 13, 25 | sstrdi 3182 | . . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ ℤ) |
27 | zdceq 9359 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 = 𝑦) | |
28 | 27 | rgen2a 2544 | . . . 4 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 |
29 | ssralv 3234 | . . . . . 6 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
30 | 29 | ralimdv 2558 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
31 | ssralv 3234 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
32 | 30, 31 | syld 45 | . . . 4 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
33 | 26, 28, 32 | mpisyl 1457 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦) |
34 | funresdfunsndc 6532 | . . 3 ⊢ ((∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦 ∧ Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) | |
35 | 33, 16, 4, 34 | syl3anc 1249 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
36 | 21, 24, 35 | 3eqtrrd 2227 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 ∖ cdif 3141 ∪ cun 3142 ⊆ wss 3144 ∅c0 3437 {csn 3607 〈cop 3610 class class class wbr 4018 dom cdm 4644 ↾ cres 4646 Fun wfun 5229 ‘cfv 5235 (class class class)co 5897 ≤ cle 8024 ℕcn 8950 ℤcz 9284 ...cfz 10040 Struct cstr 12511 ndxcnx 12512 sSet csts 12513 Slot cslot 12514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 df-uz 9560 df-fz 10041 df-struct 12517 df-slot 12519 df-sets 12522 |
This theorem is referenced by: strressid 12586 |
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