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Mirrors > Home > MPE Home > Th. List > uslgraf1oedg | Structured version Visualization version GIF version |
Description: The edge function of an undirected simple graph with loops is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) |
Ref | Expression |
---|---|
uslgraf1oedg | ⊢ (𝑉 USLGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→(𝑉Edges𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uslgraf 25874 | . 2 ⊢ (𝑉 USLGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
2 | f1f1orn 6061 | . . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸) | |
3 | edguslgra 25871 | . . . . 5 ⊢ (𝑉 USLGrph 𝐸 → (𝑉Edges𝐸) = ran 𝐸) | |
4 | 3 | eqcomd 2616 | . . . 4 ⊢ (𝑉 USLGrph 𝐸 → ran 𝐸 = (𝑉Edges𝐸)) |
5 | f1oeq3 6042 | . . . 4 ⊢ (ran 𝐸 = (𝑉Edges𝐸) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(𝑉Edges𝐸))) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑉 USLGrph 𝐸 → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(𝑉Edges𝐸))) |
7 | 2, 6 | syl5ibcom 234 | . 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (𝑉 USLGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→(𝑉Edges𝐸))) |
8 | 1, 7 | mpcom 37 | 1 ⊢ (𝑉 USLGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→(𝑉Edges𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 {crab 2900 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ran crn 5039 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ≤ cle 9954 2c2 10947 #chash 12979 USLGrph cuslg 25858 Edgescedg 25860 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-ral 2901 df-rex 2902 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 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-ov 6552 df-2nd 7060 df-uslgra 25861 df-edg 25865 |
This theorem is referenced by: (None) |
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