Theorem usgrf1oedg 26099

Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by 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 2622	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1oedg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgrf 26050	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
4		f1f1orn 6148	. . 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 25941	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
9	2	eqcomi 2631	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
10	9	rneqi 5352	. . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼
11	8, 10	syl6eq 2672	. . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
12	6, 11	syl5eq 2668	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
13	12	f1oeq3d 6134	. 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼))
14	5, 13	mpbird 247	1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {crab 2916   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  {csn 4177  dom cdm 5114  ran crn 5115  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-edg 25940  df-usgr 26046

This theorem is referenced by:  usgr2trlncl  26656