Theorem usgrf1oedg 29224

Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgrf1oedg.i	⊢ 𝐼 = (iEdg‘𝐺)
usgrf1oedg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
usgrf1oedg	⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)

Proof of Theorem usgrf1oedg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1oedg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgrf 29172	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f1orn 6859	. . 3 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
5	3, 4	syl 17	. 2 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
6		usgrf1oedg.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
7		edgval 29066	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
9	2	eqcomi 2746	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
10	9	rneqi 5948	. . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼
11	8, 10	eqtrdi 2793	. . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
12	6, 11	eqtrid 2789	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
13	12	f1oeq3d 6845	. 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼))
14	5, 13	mpbird 257	1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {crab 3436   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  ran crn 5686  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  USGraphcusgr 29166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-edg 29065  df-usgr 29168

This theorem is referenced by:  usgr2trlncl  29780