Theorem usgrf1oedg 29298

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 2741	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1oedg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgrf 29246	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f1orn 6782	. . 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 29140	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
9	2	eqcomi 2750	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
10	9	rneqi 5886	. . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼
11	8, 10	eqtrdi 2792	. . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
12	6, 11	eqtrid 2788	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
13	12	f1oeq3d 6768	. 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼))
14	5, 13	mpbird 259	1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  {crab 3393   ∖ cdif 3882  ∅c0 4264  𝒫 cpw 4532  {csn 4558  dom cdm 5621  ran crn 5622  –1-1→wf1 6486  –1-1-onto→wf1o 6488  ‘cfv 6489  2c2 12231  ♯chash 14287  Vtxcvtx 29087  iEdgciedg 29088  Edgcedg 29138  USGraphcusgr 29240

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-edg 29139  df-usgr 29242

This theorem is referenced by:  usgr2trlncl  29850