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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 873 | . . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼})) |
3 | elun 4163 | . . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼})) | |
4 | 2, 3 | sylibr 234 | . . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼})) |
5 | basprssdmsets.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
6 | snidg 4665 | . . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼}) |
8 | 7 | olcd 874 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼})) |
9 | elun 4163 | . . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼})) | |
10 | 8, 9 | sylibr 234 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼})) |
11 | 4, 10 | prssd 4827 | . 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼})) |
12 | basprssdmsets.s | . . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
13 | structex 17184 | . . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
15 | basprssdmsets.w | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
16 | setsdm 17204 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet 〈𝐼, 𝐸〉) = (dom 𝑆 ∪ {𝐼})) | |
17 | 14, 15, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → dom (𝑆 sSet 〈𝐼, 𝐸〉) = (dom 𝑆 ∪ {𝐼})) |
18 | 11, 17 | sseqtrrd 4037 | 1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet 〈𝐼, 𝐸〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 {cpr 4633 〈cop 4637 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Struct cstr 17180 sSet csts 17197 ndxcnx 17227 Basecbs 17245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-struct 17181 df-sets 17198 |
This theorem is referenced by: setsvtx 29067 setsiedg 29068 |
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