Theorem uspgrf1oedg 29107

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 2730	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1o.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uspgrf 29088	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		f1f1orn 6814	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
5	2	rneqi 5904	. . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
6		edgval 28983	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	5, 6	eqtr4i 2756	. . . 4 ⊢ ran 𝐸 = (Edg‘𝐺)
8		f1oeq3 6793	. . . 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 1540   ∈ wcel 2109  {crab 3408   ∖ cdif 3914  ∅c0 4299  𝒫 cpw 4566  {csn 4592   class class class wbr 5110  dom cdm 5641  ran crn 5642  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514   ≤ cle 11216  2c2 12248  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  Edgcedg 28981  USPGraphcuspgr 29082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-edg 28982  df-uspgr 29084

This theorem is referenced by:  uspgredgiedg  29109  uspgriedgedg  29110  uspgr2wlkeq  29581  wlkiswwlks2lem4  29809  wlkiswwlks2lem5  29810  clwlkclwwlk  29938  isuspgrim0lem  47897  upgrimwlklem2  47902  upgrimwlklem3  47903  upgrimtrlslem2  47909  upgrimtrls  47910  uspgrlimlem1  47991  uspgrlimlem2  47992  uspgrlimlem3  47993  uspgrlimlem4  47994  uspgrlim  47995