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Theorem uspgrf1oedg 29025
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 2725 . . 3 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 usgrf1o.e . . 3 𝐸 = (iEdgβ€˜πΊ)
31, 2uspgrf 29006 . 2 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
4 f1f1orn 6843 . . 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 5934 . . . . 5 ran 𝐸 = ran (iEdgβ€˜πΊ)
6 edgval 28901 . . . . 5 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
75, 6eqtr4i 2756 . . . 4 ran 𝐸 = (Edgβ€˜πΊ)
8 f1oeq3 6822 . . . 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 1533   ∈ wcel 2098  {crab 3419   βˆ– cdif 3938  βˆ…c0 4319  π’« cpw 4599  {csn 4625   class class class wbr 5144  dom cdm 5673  ran crn 5674  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543   ≀ cle 11274  2c2 12292  β™―chash 14316  Vtxcvtx 28848  iEdgciedg 28849  Edgcedg 28899  USPGraphcuspgr 29000
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 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
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 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-edg 28900  df-uspgr 29002
This theorem is referenced by:  uspgredgiedg  29027  uspgriedgedg  29028  uspgr2wlkeq  29499  wlkiswwlks2lem4  29722  wlkiswwlks2lem5  29723  clwlkclwwlk  29851  isuspgrim0lem  47277
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