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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 982 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 𝑆 ∈ 𝑉) | |
2 | opexg 4158 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | 2 | 3adant1 1000 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
4 | setsvalg 12028 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) | |
5 | 1, 3, 4 | syl2anc 409 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) |
6 | dmsnopg 5018 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
7 | 6 | difeq2d 3199 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (V ∖ dom {〈𝐴, 𝐵〉}) = (V ∖ {𝐴})) |
8 | 7 | reseq2d 4827 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) = (𝑆 ↾ (V ∖ {𝐴}))) |
9 | 8 | uneq1d 3234 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
10 | 9 | 3ad2ant3 1005 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
11 | 5, 10 | eqtrd 2173 | 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 963 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∖ cdif 3073 ∪ cun 3074 {csn 3532 〈cop 3535 dom cdm 4547 ↾ cres 4549 (class class class)co 5782 sSet csts 11996 |
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 |
This theorem depends on definitions: df-bi 116 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-ral 2422 df-rex 2423 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-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sets 12005 |
This theorem is referenced by: setsex 12030 strsetsid 12031 fvsetsid 12032 setsabsd 12037 setscom 12038 setsslid 12048 |
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