Theorem uhgredgiedgb 29158

Description: In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)

Hypothesis

Ref	Expression
uhgredgiedgb.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgredgiedgb	⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem uhgredgiedgb

Step	Hyp	Ref	Expression
1		uhgredgiedgb.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	uhgrfun 29098	. 2 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
3	1	edgiedgb 29086	. 2 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
4	2, 3	syl 17	1 ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  dom cdm 5689  Fun wfun 6557  ‘cfv 6563  iEdgciedg 29029  Edgcedg 29079  UHGraphcuhgr 29088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-edg 29080  df-uhgr 29090

This theorem is referenced by:  usgredg2vtxeuALT  29254  vtxduhgr0nedg  29525  umgr2wlk  29979  1pthon2v  30182  uhgr3cyclex  30211  isuspgrim0  47810  clnbgrgrimlem  47839  clnbgrgrim  47840  grimedg  47841