Theorem usgrf1oedg 27477

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 2738	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1oedg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgrf 27428	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f1orn 6711	. . 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 27322	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
9	2	eqcomi 2747	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
10	9	rneqi 5835	. . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼
11	8, 10	eqtrdi 2795	. . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
12	6, 11	syl5eq 2791	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
13	12	f1oeq3d 6697	. 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼))
14	5, 13	mpbird 256	1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  {csn 4558  dom cdm 5580  ran crn 5581  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  USGraphcusgr 27422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-edg 27321  df-usgr 27424

This theorem is referenced by:  usgr2trlncl  28029