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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 946 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 𝑆 ∈ 𝑉) | |
2 | opexg 4079 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | 2 | 3adant1 964 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
4 | setsvalg 11673 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) | |
5 | 1, 3, 4 | syl2anc 404 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉})) |
6 | dmsnopg 4936 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
7 | 6 | difeq2d 3133 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (V ∖ dom {〈𝐴, 𝐵〉}) = (V ∖ {𝐴})) |
8 | 7 | reseq2d 4745 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) = (𝑆 ↾ (V ∖ {𝐴}))) |
9 | 8 | uneq1d 3168 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
10 | 9 | 3ad2ant3 969 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ dom {〈𝐴, 𝐵〉})) ∪ {〈𝐴, 𝐵〉}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
11 | 5, 10 | eqtrd 2127 | 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 927 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ∖ cdif 3010 ∪ cun 3011 {csn 3466 〈cop 3469 dom cdm 4467 ↾ cres 4469 (class class class)co 5690 sSet csts 11641 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-res 4479 df-iota 5014 df-fun 5051 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-sets 11650 |
This theorem is referenced by: setsex 11675 strsetsid 11676 fvsetsid 11677 setsabsd 11682 setscom 11683 setsslid 11693 |
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