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Theorem uspgrf1oedg 26288
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 𝐸1-1-onto→(Edg‘𝐺))

Proof of Theorem uspgrf1oedg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1o.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uspgrf 26269 . 2 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 f1f1orn 6310 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
52rneqi 5507 . . . . 5 ran 𝐸 = ran (iEdg‘𝐺)
6 edgval 26161 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
75, 6eqtr4i 2785 . . . 4 ran 𝐸 = (Edg‘𝐺)
8 f1oeq3 6291 . . . 4 (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
97, 8ax-mp 5 . . 3 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
104, 9sylib 208 . 2 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
113, 10syl 17 1 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  c0 4058  𝒫 cpw 4302  {csn 4321   class class class wbr 4804  dom cdm 5266  ran crn 5267  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049  cle 10287  2c2 11282  chash 13331  Vtxcvtx 26094  iEdgciedg 26095  Edgcedg 26159  USPGraphcuspgr 26263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-edg 26160  df-uspgr 26265
This theorem is referenced by:  uspgr2wlkeq  26773  wlkiswwlks2lem4  27002  wlkiswwlks2lem5  27003  clwlkclwwlk  27146
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