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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 12010 | . . . 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 12012 | . . . . . . . . 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 996 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 ⊆ (𝑀...𝑁)) |
9 | 7 | simp1d 994 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) |
10 | 9 | simp1d 994 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | fzssnn 9879 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | |
12 | 10, 11 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀...𝑁) ⊆ ℕ) |
13 | 8, 12 | sstrd 3112 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 ⊆ ℕ) |
14 | 13, 4 | sseldd 3103 | . . . . 5 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
15 | 5, 3, 14 | strnfvnd 12018 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
16 | strsetsid.f | . . . . 5 ⊢ (𝜑 → Fun 𝑆) | |
17 | funfvex 5446 | . . . . 5 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → (𝑆‘(𝐸‘ndx)) ∈ V) | |
18 | 16, 4, 17 | syl2anc 409 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) ∈ V) |
19 | 15, 18 | eqeltrd 2217 | . . 3 ⊢ (𝜑 → (𝐸‘𝑆) ∈ V) |
20 | setsvala 12029 | . . 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 1217 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
22 | 15 | opeq2d 3720 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
23 | 22 | sneqd 3545 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
24 | 23 | uneq2d 3235 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
25 | nnssz 9095 | . . . . 5 ⊢ ℕ ⊆ ℤ | |
26 | 13, 25 | sstrdi 3114 | . . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ ℤ) |
27 | zdceq 9150 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 = 𝑦) | |
28 | 27 | rgen2a 2489 | . . . 4 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 |
29 | ssralv 3166 | . . . . . 6 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
30 | 29 | ralimdv 2503 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
31 | ssralv 3166 | . . . . 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 1423 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦) |
34 | funresdfunsndc 6410 | . . 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 1217 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
36 | 21, 24, 35 | 3eqtrrd 2178 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 820 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ∖ cdif 3073 ∪ cun 3074 ⊆ wss 3076 ∅c0 3368 {csn 3532 〈cop 3535 class class class wbr 3937 dom cdm 4547 ↾ cres 4549 Fun wfun 5125 ‘cfv 5131 (class class class)co 5782 ≤ cle 7825 ℕcn 8744 ℤcz 9078 ...cfz 9821 Struct cstr 11994 ndxcnx 11995 sSet csts 11996 Slot cslot 11997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-struct 12000 df-slot 12002 df-sets 12005 |
This theorem is referenced by: (None) |
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