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Theorem uspgrf1oedg 28960
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 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
uspgrf1oedg (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))

Proof of Theorem uspgrf1oedg
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2727 . . 3 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 usgrf1o.e . . 3 𝐸 = (iEdgβ€˜πΊ)
31, 2uspgrf 28941 . 2 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
4 f1f1orn 6844 . . 3 (𝐸:dom 𝐸–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2} β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
52rneqi 5933 . . . . 5 ran 𝐸 = ran (iEdgβ€˜πΊ)
6 edgval 28836 . . . . 5 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
75, 6eqtr4i 2758 . . . 4 ran 𝐸 = (Edgβ€˜πΊ)
8 f1oeq3 6823 . . . 4 (ran 𝐸 = (Edgβ€˜πΊ) β†’ (𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ)))
97, 8ax-mp 5 . . 3 (𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
104, 9sylib 217 . 2 (𝐸:dom 𝐸–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2} β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
113, 10syl 17 1 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  {crab 3427   βˆ– cdif 3941  βˆ…c0 4318  π’« cpw 4598  {csn 4624   class class class wbr 5142  dom cdm 5672  ran crn 5673  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542   ≀ cle 11265  2c2 12283  β™―chash 14307  Vtxcvtx 28783  iEdgciedg 28784  Edgcedg 28834  USPGraphcuspgr 28935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  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 28835  df-uspgr 28937
This theorem is referenced by:  uspgredgiedg  28962  uspgriedgedg  28963  uspgr2wlkeq  29434  wlkiswwlks2lem4  29657  wlkiswwlks2lem5  29658  clwlkclwwlk  29786  isuspgrim0lem  47082
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