![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > edg0iedg0 | Structured version Visualization version GIF version |
Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.) |
Ref | Expression |
---|---|
edg0iedg0.i | ⊢ 𝐼 = (iEdg‘𝐺) |
edg0iedg0.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
edg0iedg0 | ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edg0iedg0.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | edgval 28174 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | 1, 2 | eqtri 2759 | . . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺) |
4 | 3 | eqeq1i 2736 | . . 3 ⊢ (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅) |
5 | 4 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ ran (iEdg‘𝐺) = ∅)) |
6 | edg0iedg0.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | eqcomi 2740 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5928 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 8 | eqeq1i 2736 | . . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅) |
10 | 9 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) |
11 | funrel 6554 | . . 3 ⊢ (Fun 𝐼 → Rel 𝐼) | |
12 | relrn0 5960 | . . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅)) | |
13 | 12 | bicomd 222 | . . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
14 | 11, 13 | syl 17 | . 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
15 | 5, 10, 14 | 3bitrd 304 | 1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∅c0 4318 ran crn 5670 Rel wrel 5674 Fun wfun 6526 ‘cfv 6532 iEdgciedg 28122 Edgcedg 28172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fv 6540 df-edg 28173 |
This theorem is referenced by: uhgriedg0edg0 28252 egrsubgr 28399 vtxduhgr0e 28600 |
Copyright terms: Public domain | W3C validator |