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Mirrors > Home > ILE Home > Th. List > setsvala | GIF version |
Description: Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.) |
Ref | Expression |
---|---|
setsvala | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 999 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 𝑆 ∈ 𝑉) | |
2 | opexg 4243 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V) | |
3 | 2 | 3adant1 1017 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V) |
4 | setsvalg 12516 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐴, 𝐵⟩})) ∪ {⟨𝐴, 𝐵⟩})) | |
5 | 1, 3, 4 | syl2anc 411 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ dom {⟨𝐴, 𝐵⟩})) ∪ {⟨𝐴, 𝐵⟩})) |
6 | dmsnopg 5115 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴}) | |
7 | 6 | difeq2d 3268 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (V ∖ dom {⟨𝐴, 𝐵⟩}) = (V ∖ {𝐴})) |
8 | 7 | reseq2d 4922 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑆 ↾ (V ∖ dom {⟨𝐴, 𝐵⟩})) = (𝑆 ↾ (V ∖ {𝐴}))) |
9 | 8 | uneq1d 3303 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝑆 ↾ (V ∖ dom {⟨𝐴, 𝐵⟩})) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) |
10 | 9 | 3ad2ant3 1022 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ dom {⟨𝐴, 𝐵⟩})) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) |
11 | 5, 10 | eqtrd 2222 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∖ cdif 3141 ∪ cun 3142 {csn 3607 ⟨cop 3610 dom cdm 4641 ↾ cres 4643 (class class class)co 5891 sSet csts 12484 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 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-ral 2473 df-rex 2474 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-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-res 4653 df-iota 5193 df-fun 5233 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-sets 12493 |
This theorem is referenced by: setsex 12518 strsetsid 12519 fvsetsid 12520 setsabsd 12525 setscom 12526 setsslid 12537 |
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