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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 27094 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | 1, 2 | eqtri 2759 | . . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺) |
4 | 3 | eqeq1i 2741 | . . 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 2745 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
8 | 7 | rneqi 5791 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
9 | 8 | eqeq1i 2741 | . . 3 ⊢ (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅) |
10 | 9 | a1i 11 | . 2 ⊢ (Fun 𝐼 → (ran (iEdg‘𝐺) = ∅ ↔ ran 𝐼 = ∅)) |
11 | funrel 6375 | . . 3 ⊢ (Fun 𝐼 → Rel 𝐼) | |
12 | relrn0 5823 | . . . 4 ⊢ (Rel 𝐼 → (𝐼 = ∅ ↔ ran 𝐼 = ∅)) | |
13 | 12 | bicomd 226 | . . 3 ⊢ (Rel 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
14 | 11, 13 | syl 17 | . 2 ⊢ (Fun 𝐼 → (ran 𝐼 = ∅ ↔ 𝐼 = ∅)) |
15 | 5, 10, 14 | 3bitrd 308 | 1 ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∅c0 4223 ran crn 5537 Rel wrel 5541 Fun wfun 6352 ‘cfv 6358 iEdgciedg 27042 Edgcedg 27092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-edg 27093 |
This theorem is referenced by: uhgriedg0edg0 27172 egrsubgr 27319 vtxduhgr0e 27520 |
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