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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 29084 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | 1, 2 | eqtri 2768 | . . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺) |
4 | 3 | eqeq1i 2745 | . . 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 2749 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5962 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 8 | eqeq1i 2745 | . . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅) |
10 | 9 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) |
11 | funrel 6595 | . . 3 ⊢ (Fun 𝐼 → Rel 𝐼) | |
12 | relrn0 5995 | . . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅)) | |
13 | 12 | bicomd 223 | . . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
14 | 11, 13 | syl 17 | . 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
15 | 5, 10, 14 | 3bitrd 305 | 1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∅c0 4352 ran crn 5701 Rel wrel 5705 Fun wfun 6567 ‘cfv 6573 iEdgciedg 29032 Edgcedg 29082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-edg 29083 |
This theorem is referenced by: uhgriedg0edg0 29162 egrsubgr 29312 vtxduhgr0e 29514 |
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