Theorem uspgrf1oedg 29151

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 𝐸–1-1-onto→(Edg‘𝐺))

Proof of Theorem uspgrf1oedg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1o.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uspgrf 29132	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		f1f1orn 6774	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
5	2	rneqi 5876	. . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
6		edgval 29027	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	5, 6	eqtr4i 2757	. . . 4 ⊢ ran 𝐸 = (Edg‘𝐺)
8		f1oeq3 6753	. . . 4 ⊢ (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)))
9	7, 8	ax-mp 5	. . 3 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))
10	4, 9	sylib 218	. 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))
11	3, 10	syl 17	1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {crab 3395   ∖ cdif 3894  ∅c0 4280  𝒫 cpw 4547  {csn 4573   class class class wbr 5089  dom cdm 5614  ran crn 5615  –1-1→wf1 6478  –1-1-onto→wf1o 6480  ‘cfv 6481   ≤ cle 11147  2c2 12180  ♯chash 14237  Vtxcvtx 28974  iEdgciedg 28975  Edgcedg 29025  USPGraphcuspgr 29126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-edg 29026  df-uspgr 29128

This theorem is referenced by:  uspgredgiedg  29153  uspgriedgedg  29154  uspgr2wlkeq  29624  wlkiswwlks2lem4  29850  wlkiswwlks2lem5  29851  clwlkclwwlk  29982  isuspgrim0lem  48003  upgrimwlklem2  48008  upgrimwlklem3  48009  upgrimtrlslem2  48015  upgrimtrls  48016  uspgrlimlem1  48098  uspgrlimlem2  48099  uspgrlimlem3  48100  uspgrlimlem4  48101  uspgrlim  48102