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Mirrors > Home > MPE Home > Th. List > setsiedg | Structured version Visualization version GIF version |
Description: The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (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 |
---|---|
setsiedg | ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsvtx.s | . . . 4 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
2 | fvexd 6900 | . . . 4 ⊢ (𝜑 → (.ef‘ndx) ∈ V) | |
3 | setsvtx.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
4 | 1, 2, 3 | setsn0fun 17115 | . . 3 ⊢ (𝜑 → Fun ((𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩) ∖ {∅})) |
5 | setsvtx.b | . . . 4 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
6 | 1, 2, 3, 5 | basprssdmsets 17166 | . . 3 ⊢ (𝜑 → {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)) |
7 | funiedgval 28787 | . . 3 ⊢ ((Fun ((𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)) → (iEdg‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)) = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))) | |
8 | 4, 6, 7 | syl2anc 583 | . 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)) = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))) |
9 | setsvtx.i | . . . . . 6 ⊢ 𝐼 = (.ef‘ndx) | |
10 | 9 | opeq1i 4871 | . . . . 5 ⊢ ⟨𝐼, 𝐸⟩ = ⟨(.ef‘ndx), 𝐸⟩ |
11 | 10 | oveq2i 7416 | . . . 4 ⊢ (𝐺 sSet ⟨𝐼, 𝐸⟩) = (𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩) |
12 | 11 | fveq2i 6888 | . . 3 ⊢ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (iEdg‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)) |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (iEdg‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))) |
14 | structex 17092 | . . . 4 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V) | |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
16 | edgfid 28756 | . . . 4 ⊢ .ef = Slot (.ef‘ndx) | |
17 | 16 | setsid 17150 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐸 ∈ 𝑊) → 𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))) |
18 | 15, 3, 17 | syl2anc 583 | . 2 ⊢ (𝜑 → 𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))) |
19 | 8, 13, 18 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 {csn 4623 {cpr 4625 ⟨cop 4629 class class class wbr 5141 dom cdm 5669 Fun wfun 6531 ‘cfv 6537 (class class class)co 7405 Struct cstr 17088 sSet csts 17105 ndxcnx 17135 Basecbs 17153 .efcedgf 28754 iEdgciedg 28765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-edgf 28755 df-iedg 28767 |
This theorem is referenced by: uhgrstrrepe 28846 usgrstrrepe 29000 structtocusgr 29211 |
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