Theorem uspgrf1oedg 26952

Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)

Hypothesis

Ref	Expression
usgrf1o.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
uspgrf1oedg	⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))

Proof of Theorem uspgrf1oedg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1o.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uspgrf 26933	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		f1f1orn 6621	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
5	2	rneqi 5802	. . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
6		edgval 26828	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	5, 6	eqtr4i 2847	. . . 4 ⊢ ran 𝐸 = (Edg‘𝐺)
8		f1oeq3 6601	. . . 4 ⊢ (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)))
9	7, 8	ax-mp 5	. . 3 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))
10	4, 9	sylib 220	. 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))
11	3, 10	syl 17	1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {crab 3142   ∖ cdif 3933  ∅c0 4291  𝒫 cpw 4539  {csn 4561   class class class wbr 5059  dom cdm 5550  ran crn 5551  –1-1→wf1 6347  –1-1-onto→wf1o 6349  ‘cfv 6350   ≤ cle 10670  2c2 11686  ♯chash 13684  Vtxcvtx 26775  iEdgciedg 26776  Edgcedg 26826  USPGraphcuspgr 26927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-edg 26827  df-uspgr 26929

This theorem is referenced by:  uspgr2wlkeq  27421  wlkiswwlks2lem4  27644  wlkiswwlks2lem5  27645  clwlkclwwlk  27774