![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ (π β π Struct π) |
basprssdmsets.i | β’ (π β πΌ β π) |
basprssdmsets.w | β’ (π β πΈ β π) |
basprssdmsets.b | β’ (π β (Baseβndx) β dom π) |
Ref | Expression |
---|---|
basprssdmsets | β’ (π β {(Baseβndx), πΌ} β dom (π sSet β¨πΌ, πΈβ©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basprssdmsets.b | . . . . 5 β’ (π β (Baseβndx) β dom π) | |
2 | 1 | orcd 871 | . . . 4 β’ (π β ((Baseβndx) β dom π β¨ (Baseβndx) β {πΌ})) |
3 | elun 4148 | . . . 4 β’ ((Baseβndx) β (dom π βͺ {πΌ}) β ((Baseβndx) β dom π β¨ (Baseβndx) β {πΌ})) | |
4 | 2, 3 | sylibr 233 | . . 3 β’ (π β (Baseβndx) β (dom π βͺ {πΌ})) |
5 | basprssdmsets.i | . . . . . 6 β’ (π β πΌ β π) | |
6 | snidg 4662 | . . . . . 6 β’ (πΌ β π β πΌ β {πΌ}) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β πΌ β {πΌ}) |
8 | 7 | olcd 872 | . . . 4 β’ (π β (πΌ β dom π β¨ πΌ β {πΌ})) |
9 | elun 4148 | . . . 4 β’ (πΌ β (dom π βͺ {πΌ}) β (πΌ β dom π β¨ πΌ β {πΌ})) | |
10 | 8, 9 | sylibr 233 | . . 3 β’ (π β πΌ β (dom π βͺ {πΌ})) |
11 | 4, 10 | prssd 4825 | . 2 β’ (π β {(Baseβndx), πΌ} β (dom π βͺ {πΌ})) |
12 | basprssdmsets.s | . . . 4 β’ (π β π Struct π) | |
13 | structex 17085 | . . . 4 β’ (π Struct π β π β V) | |
14 | 12, 13 | syl 17 | . . 3 β’ (π β π β V) |
15 | basprssdmsets.w | . . 3 β’ (π β πΈ β π) | |
16 | setsdm 17105 | . . 3 β’ ((π β V β§ πΈ β π) β dom (π sSet β¨πΌ, πΈβ©) = (dom π βͺ {πΌ})) | |
17 | 14, 15, 16 | syl2anc 584 | . 2 β’ (π β dom (π sSet β¨πΌ, πΈβ©) = (dom π βͺ {πΌ})) |
18 | 11, 17 | sseqtrrd 4023 | 1 β’ (π β {(Baseβndx), πΌ} β dom (π sSet β¨πΌ, πΈβ©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 845 = wceq 1541 β wcel 2106 Vcvv 3474 βͺ cun 3946 β wss 3948 {csn 4628 {cpr 4630 β¨cop 4634 class class class wbr 5148 dom cdm 5676 βcfv 6543 (class class class)co 7411 Struct cstr 17081 sSet csts 17098 ndxcnx 17128 Basecbs 17146 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-struct 17082 df-sets 17099 |
This theorem is referenced by: setsvtx 28333 setsiedg 28334 |
Copyright terms: Public domain | W3C validator |