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Mirrors > Home > MPE Home > Th. List > edgfiedgval | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
basvtxval.s | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
basvtxval.d | ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) |
edgfiedgval.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
edgfiedgval.f | ⊢ (𝜑 → 〈(.ef‘ndx), 𝐸〉 ∈ 𝐺) |
Ref | Expression |
---|---|
edgfiedgval | ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basvtxval.s | . . . 4 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
2 | structn0fun 16495 | . . . 4 ⊢ (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅})) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Fun (𝐺 ∖ {∅})) |
4 | basvtxval.d | . . 3 ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) | |
5 | funiedgdmge2val 26797 | . . 3 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
6 | 3, 4, 5 | syl2anc 586 | . 2 ⊢ (𝜑 → (iEdg‘𝐺) = (.ef‘𝐺)) |
7 | edgfid 26776 | . . 3 ⊢ .ef = Slot (.ef‘ndx) | |
8 | structex 16494 | . . . 4 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V) | |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
10 | structfung 16498 | . . . 4 ⊢ (𝐺 Struct 𝑋 → Fun ◡◡𝐺) | |
11 | 1, 10 | syl 17 | . . 3 ⊢ (𝜑 → Fun ◡◡𝐺) |
12 | edgfiedgval.f | . . 3 ⊢ (𝜑 → 〈(.ef‘ndx), 𝐸〉 ∈ 𝐺) | |
13 | edgfiedgval.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
14 | 7, 9, 11, 12, 13 | strfv2d 16529 | . 2 ⊢ (𝜑 → 𝐸 = (.ef‘𝐺)) |
15 | 6, 14 | eqtr4d 2859 | 1 ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 {csn 4567 〈cop 4573 class class class wbr 5066 ◡ccnv 5554 dom cdm 5555 Fun wfun 6349 ‘cfv 6355 ≤ cle 10676 2c2 11693 ♯chash 13691 Struct cstr 16479 ndxcnx 16480 .efcedgf 26774 iEdgciedg 26782 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-edgf 26775 df-iedg 26784 |
This theorem is referenced by: structgrssiedg 26810 |
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