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Theorem usgrf1oedg 28197
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 2733 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
31, 2usgrf 28148 . . 3 (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) = 2})
4 f1f1orn 6796 . . 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 28042 . . . . . 6 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph β†’ (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ))
92eqcomi 2742 . . . . . 6 (iEdgβ€˜πΊ) = 𝐼
109rneqi 5893 . . . . 5 ran (iEdgβ€˜πΊ) = ran 𝐼
118, 10eqtrdi 2789 . . . 4 (𝐺 ∈ USGraph β†’ (Edgβ€˜πΊ) = ran 𝐼)
126, 11eqtrid 2785 . . 3 (𝐺 ∈ USGraph β†’ 𝐸 = ran 𝐼)
1312f1oeq3d 6782 . 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 1542   ∈ wcel 2107  {crab 3406   βˆ– cdif 3908  βˆ…c0 4283  π’« cpw 4561  {csn 4587  dom cdm 5634  ran crn 5635  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  2c2 12213  β™―chash 14236  Vtxcvtx 27989  iEdgciedg 27990  Edgcedg 28040  USGraphcusgr 28142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-edg 28041  df-usgr 28144
This theorem is referenced by:  usgr2trlncl  28750
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