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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 | 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 870 | . . . 4 β’ (π β ((Baseβndx) β dom π β¨ (Baseβndx) β {πΌ})) |
3 | elun 4149 | . . . 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 4663 | . . . . . 6 β’ (πΌ β π β πΌ β {πΌ}) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β πΌ β {πΌ}) |
8 | 7 | olcd 871 | . . . 4 β’ (π β (πΌ β dom π β¨ πΌ β {πΌ})) |
9 | elun 4149 | . . . 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 4826 | . 2 β’ (π β {(Baseβndx), πΌ} β (dom π βͺ {πΌ})) |
12 | basprssdmsets.s | . . . 4 β’ (π β π Struct π) | |
13 | structex 17088 | . . . 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 17108 | . . 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 583 | . 2 β’ (π β dom (π sSet β¨πΌ, πΈβ©) = (dom π βͺ {πΌ})) |
18 | 11, 17 | sseqtrrd 4024 | 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 844 = wceq 1540 β wcel 2105 Vcvv 3473 βͺ cun 3947 β wss 3949 {csn 4629 {cpr 4631 β¨cop 4635 class class class wbr 5149 dom cdm 5677 βcfv 6544 (class class class)co 7412 Struct cstr 17084 sSet csts 17101 ndxcnx 17131 Basecbs 17149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-res 5689 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-struct 17085 df-sets 17102 |
This theorem is referenced by: setsvtx 28559 setsiedg 28560 |
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