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Mirrors > Home > MPE Home > Th. List > setsvtx | Structured version Visualization version GIF version |
Description: The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
setsvtx.i | ⊢ 𝐼 = (.ef‘ndx) |
setsvtx.s | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
setsvtx.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) |
setsvtx.e | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
Ref | Expression |
---|---|
setsvtx | ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsvtx.s | . . . 4 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
2 | setsvtx.i | . . . . . 6 ⊢ 𝐼 = (.ef‘ndx) | |
3 | 2 | fvexi 6915 | . . . . 5 ⊢ 𝐼 ∈ V |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
5 | setsvtx.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
6 | 1, 4, 5 | setsn0fun 17175 | . . 3 ⊢ (𝜑 → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅})) |
7 | 2 | eqcomi 2735 | . . . . 5 ⊢ (.ef‘ndx) = 𝐼 |
8 | 7 | preq2i 4746 | . . . 4 ⊢ {(Base‘ndx), (.ef‘ndx)} = {(Base‘ndx), 𝐼} |
9 | setsvtx.b | . . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
10 | 1, 4, 5, 9 | basprssdmsets 17226 | . . . 4 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉)) |
11 | 8, 10 | eqsstrid 4028 | . . 3 ⊢ (𝜑 → {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉)) |
12 | funvtxval 28954 | . . 3 ⊢ ((Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉)) → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘(𝐺 sSet 〈𝐼, 𝐸〉))) | |
13 | 6, 11, 12 | syl2anc 582 | . 2 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘(𝐺 sSet 〈𝐼, 𝐸〉))) |
14 | baseid 17216 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
15 | basendxnedgfndx 28931 | . . . 4 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
16 | 15, 2 | neeqtrri 3004 | . . 3 ⊢ (Base‘ndx) ≠ 𝐼 |
17 | 14, 16 | setsnid 17211 | . 2 ⊢ (Base‘𝐺) = (Base‘(𝐺 sSet 〈𝐼, 𝐸〉)) |
18 | 13, 17 | eqtr4di 2784 | 1 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 {csn 4633 {cpr 4635 〈cop 4639 class class class wbr 5153 dom cdm 5682 Fun wfun 6548 ‘cfv 6554 (class class class)co 7424 Struct cstr 17148 sSet csts 17165 ndxcnx 17195 Basecbs 17213 .efcedgf 28922 Vtxcvtx 28932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-edgf 28923 df-vtx 28934 |
This theorem is referenced by: uhgrstrrepe 29014 usgrstrrepe 29171 structtocusgr 29382 |
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