Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > struct2griedg | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 23-Sep-2020.) |
Ref | Expression |
---|---|
struct2grvtx.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} |
Ref | Expression |
---|---|
struct2griedg | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgfndxnn 25669 | . . . 4 ⊢ (.ef‘ndx) ∈ ℕ | |
2 | baseltedgf 25671 | . . . 4 ⊢ (Base‘ndx) < (.ef‘ndx) | |
3 | struct2grvtx.g | . . . 4 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} | |
4 | 1, 2, 3 | structvtxvallem 25697 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐺 ∈ V ∧ Fun 𝐺 ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺)) |
5 | fundif 5849 | . . . 4 ⊢ (Fun 𝐺 → Fun (𝐺 ∖ {∅})) | |
6 | 5 | 3anim2i 1242 | . . 3 ⊢ ((𝐺 ∈ V ∧ Fun 𝐺 ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺)) |
7 | funiedgval 25696 | . . 3 ⊢ ((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 4, 6, 7 | 3syl 18 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (.ef‘𝐺)) |
9 | df-edgf 25668 | . . . . . 6 ⊢ .ef = Slot ;18 | |
10 | 1nn0 11185 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
11 | 8nn 11068 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
12 | 10, 11 | decnncl 11394 | . . . . . 6 ⊢ ;18 ∈ ℕ |
13 | 9, 12 | ndxid 15716 | . . . . 5 ⊢ .ef = Slot (.ef‘ndx) |
14 | 3, 2, 1, 13 | 2strop1 15814 | . . . 4 ⊢ (𝐸 ∈ 𝑌 → 𝐸 = (.ef‘𝐺)) |
15 | 14 | eqcomd 2616 | . . 3 ⊢ (𝐸 ∈ 𝑌 → (.ef‘𝐺) = 𝐸) |
16 | 15 | adantl 481 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (.ef‘𝐺) = 𝐸) |
17 | 8, 16 | eqtrd 2644 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 〈cop 4131 dom cdm 5038 Fun wfun 5798 ‘cfv 5804 1c1 9816 8c8 10953 ;cdc 11369 ndxcnx 15692 Basecbs 15695 .efcedgf 25667 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-edgf 25668 df-iedg 25676 |
This theorem is referenced by: grastruct 25707 edgastruct 25797 |
Copyright terms: Public domain | W3C validator |