Theorem uhgredgiedgb 29734

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 29674	. 2 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
3	1	edgiedgb 29662	. 2 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
4	2, 3	syl 18	1 ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  dom cdm 5655  Fun wfun 6552  ‘cfv 6558  iEdgciedg 29605  Edgcedg 29655  UHGraphcuhgr 29664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-edg 29656  df-uhgr 29666

This theorem is used by:  usgredg2vtxeuALT  29833  vtxduhgr0nedg  30103  umgr2wlk  30568  1pthon2v  30784  uhgr3cyclex  30813  isuspgrim0  49197  clnbgrgrimlem  49236  clnbgrgrim  49237  grimedg  49238