Theorem usgrf1oedg 29062

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 2725	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1oedg.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	usgrf 29010	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f1orn 6844	. . 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 28904	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
9	2	eqcomi 2734	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
10	9	rneqi 5933	. . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼
11	8, 10	eqtrdi 2781	. . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
12	6, 11	eqtrid 2777	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
13	12	f1oeq3d 6830	. 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 1533   ∈ wcel 2098  {crab 3419   ∖ cdif 3937  ∅c0 4318  𝒫 cpw 4598  {csn 4624  dom cdm 5672  ran crn 5673  –1-1→wf1 6539  –1-1-onto→wf1o 6541  ‘cfv 6542  2c2 12295  ♯chash 14319  Vtxcvtx 28851  iEdgciedg 28852  Edgcedg 28902  USGraphcusgr 29004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-edg 28903  df-usgr 29006

This theorem is referenced by:  usgr2trlncl  29616