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Mirrors > Home > MPE Home > Th. List > basprssdmsets | Structured version Visualization version GIF version |
Description: The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
basprssdmsets.s | ⊢ (𝜑 → 𝑆 Struct 𝑋) |
basprssdmsets.i | ⊢ (𝜑 → 𝐼 ∈ 𝑈) |
basprssdmsets.w | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
basprssdmsets.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) |
Ref | Expression |
---|---|
basprssdmsets | ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet 〈𝐼, 𝐸〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basprssdmsets.b | . . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) | |
2 | 1 | orcd 870 | . . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼})) |
3 | elun 4076 | . . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼})) | |
4 | 2, 3 | sylibr 237 | . . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼})) |
5 | basprssdmsets.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
6 | snidg 4559 | . . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼}) |
8 | 7 | olcd 871 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼})) |
9 | elun 4076 | . . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼})) | |
10 | 8, 9 | sylibr 237 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼})) |
11 | 4, 10 | prssd 4715 | . 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼})) |
12 | basprssdmsets.s | . . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
13 | structex 16486 | . . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
15 | basprssdmsets.w | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
16 | setsdm 16509 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet 〈𝐼, 𝐸〉) = (dom 𝑆 ∪ {𝐼})) | |
17 | 14, 15, 16 | syl2anc 587 | . 2 ⊢ (𝜑 → dom (𝑆 sSet 〈𝐼, 𝐸〉) = (dom 𝑆 ∪ {𝐼})) |
18 | 11, 17 | sseqtrrd 3956 | 1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet 〈𝐼, 𝐸〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 {csn 4525 {cpr 4527 〈cop 4531 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Struct cstr 16471 ndxcnx 16472 sSet csts 16473 Basecbs 16475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-struct 16477 df-sets 16482 |
This theorem is referenced by: setsvtx 26828 setsiedg 26829 |
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