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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 874 | . . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼})) | 
| 3 | elun 4153 | . . . 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 4660 | . . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼}) | |
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼}) | 
| 8 | 7 | olcd 875 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼})) | 
| 9 | elun 4153 | . . . 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 4822 | . 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼})) | 
| 12 | basprssdmsets.s | . . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋) | |
| 13 | structex 17187 | . . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
| 14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) | 
| 15 | basprssdmsets.w | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
| 16 | setsdm 17207 | . . 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 4021 | 1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet 〈𝐼, 𝐸〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 〈cop 4632 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Struct cstr 17183 sSet csts 17200 ndxcnx 17230 Basecbs 17247 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-struct 17184 df-sets 17201 | 
| This theorem is referenced by: setsvtx 29052 setsiedg 29053 | 
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