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Mirrors > Home > ILE Home > Th. List > setsex | GIF version |
Description: Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.) |
Ref | Expression |
---|---|
setsex | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsvala 11917 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
2 | resexg 4829 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾ (V ∖ {𝐴})) ∈ V) | |
3 | 2 | 3ad2ant1 987 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 ↾ (V ∖ {𝐴})) ∈ V) |
4 | opexg 4120 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
5 | 4 | 3adant1 984 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) |
6 | snexg 4078 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ V → {〈𝐴, 𝐵〉} ∈ V) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} ∈ V) |
8 | unexg 4334 | . . 3 ⊢ (((𝑆 ↾ (V ∖ {𝐴})) ∈ V ∧ {〈𝐴, 𝐵〉} ∈ V) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) | |
9 | 3, 7, 8 | syl2anc 408 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) |
10 | 1, 9 | eqeltrd 2194 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 947 ∈ wcel 1465 Vcvv 2660 ∖ cdif 3038 ∪ cun 3039 {csn 3497 〈cop 3500 ↾ cres 4511 (class class class)co 5742 sSet csts 11884 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sets 11893 |
This theorem is referenced by: setsabsd 11925 setscom 11926 setsslnid 11937 ressval2 11946 |
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