Theorem uhgredgiedgb 29182

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 29122	. 2 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
3	1	edgiedgb 29110	. 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 1542   ∈ wcel 2114  ∃wrex 3061  dom cdm 5625  Fun wfun 6487  ‘cfv 6493  iEdgciedg 29053  Edgcedg 29103  UHGraphcuhgr 29112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-edg 29104  df-uhgr 29114

This theorem is referenced by:  usgredg2vtxeuALT  29278  vtxduhgr0nedg  29549  umgr2wlk  30005  1pthon2v  30211  uhgr3cyclex  30240  isuspgrim0  48176  clnbgrgrimlem  48215  clnbgrgrim  48216  grimedg  48217