Theorem uhgredgiedgb 29161

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 29101	. 2 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
3	1	edgiedgb 29089	. 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 2108  ∃wrex 3076  dom cdm 5700  Fun wfun 6567  ‘cfv 6573  iEdgciedg 29032  Edgcedg 29082  UHGraphcuhgr 29091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-edg 29083  df-uhgr 29093

This theorem is referenced by:  usgredg2vtxeuALT  29257  vtxduhgr0nedg  29528  umgr2wlk  29982  1pthon2v  30185  uhgr3cyclex  30214  isuspgrim0  47756  clnbgrgrimlem  47785  clnbgrgrim  47786  grimedg  47787