Theorem uspgrf1oedg 29523

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 2763	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgrf1o.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uspgrf 29504	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4		f1f1orn 6832	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
5	2	rneqi 5927	. . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
6		edgval 29399	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	5, 6	eqtr4i 2789	. . . 4 ⊢ ran 𝐸 = (Edg‘𝐺)
8		f1oeq3 6810	. . . 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 221	. 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))
11	3, 10	syl 18	1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562  {csn 4589   class class class wbr 5109  dom cdm 5661  ran crn 5662  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536   ≤ cle 11239  2c2 12290  ♯chash 14362  Vtxcvtx 29346  iEdgciedg 29347  Edgcedg 29397  USPGraphcuspgr 29498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-edg 29398  df-uspgr 29500

This theorem is referenced by:  uspgredgiedg  29525  uspgriedgedg  29526  uspgr2wlkeq  29995  wlkiswwlks2lem4  30221  wlkiswwlks2lem5  30222  clwlkclwwlk  30353  isuspgrim0lem  48678  upgrimwlklem2  48683  upgrimwlklem3  48684  upgrimtrlslem2  48690  upgrimtrls  48691  uspgrlimlem1  48773  uspgrlimlem2  48774  uspgrlimlem3  48775  uspgrlimlem4  48776  uspgrlim  48777