MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgrf1oedg Structured version   Visualization version   GIF version

Theorem usgrf1oedg 28972
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 = (iEdgβ€˜πΊ)
usgrf1oedg.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
usgrf1oedg (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1-onto→𝐸)

Proof of Theorem usgrf1oedg
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
31, 2usgrf 28923 . . 3 (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) = 2})
4 f1f1orn 6838 . . 3 (𝐼:dom 𝐼–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) = 2} β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
53, 4syl 17 . 2 (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
6 usgrf1oedg.e . . . 4 𝐸 = (Edgβ€˜πΊ)
7 edgval 28817 . . . . . 6 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph β†’ (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ))
92eqcomi 2735 . . . . . 6 (iEdgβ€˜πΊ) = 𝐼
109rneqi 5930 . . . . 5 ran (iEdgβ€˜πΊ) = ran 𝐼
118, 10eqtrdi 2782 . . . 4 (𝐺 ∈ USGraph β†’ (Edgβ€˜πΊ) = ran 𝐼)
126, 11eqtrid 2778 . . 3 (𝐺 ∈ USGraph β†’ 𝐸 = ran 𝐼)
1312f1oeq3d 6824 . 2 (𝐺 ∈ USGraph β†’ (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼))
145, 13mpbird 257 1 (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1-onto→𝐸)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426   βˆ– cdif 3940  βˆ…c0 4317  π’« cpw 4597  {csn 4623  dom cdm 5669  ran crn 5670  β€“1-1β†’wf1 6534  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  2c2 12271  β™―chash 14295  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  USGraphcusgr 28917
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-edg 28816  df-usgr 28919
This theorem is referenced by:  usgr2trlncl  29526
  Copyright terms: Public domain W3C validator