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Theorem issubgr2 26970
Description: The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.)
Hypotheses
Ref Expression
issubgr.v 𝑉 = (Vtx‘𝑆)
issubgr.a 𝐴 = (Vtx‘𝐺)
issubgr.i 𝐼 = (iEdg‘𝑆)
issubgr.b 𝐵 = (iEdg‘𝐺)
issubgr.e 𝐸 = (Edg‘𝑆)
Assertion
Ref Expression
issubgr2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))

Proof of Theorem issubgr2
StepHypRef Expression
1 issubgr.v . . . 4 𝑉 = (Vtx‘𝑆)
2 issubgr.a . . . 4 𝐴 = (Vtx‘𝐺)
3 issubgr.i . . . 4 𝐼 = (iEdg‘𝑆)
4 issubgr.b . . . 4 𝐵 = (iEdg‘𝐺)
5 issubgr.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5issubgr 26969 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
763adant2 1125 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8 resss 5876 . . . . 5 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9 sseq1 3995 . . . . 5 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
108, 9mpbiri 259 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
11 funssres 6394 . . . . . . 7 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1211eqcomd 2831 . . . . . 6 ((Fun 𝐵𝐼𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
1312ex 413 . . . . 5 (Fun 𝐵 → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
14133ad2ant2 1128 . . . 4 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
1510, 14impbid2 227 . . 3 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼𝐵))
16153anbi2d 1434 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → ((𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
177, 16bitrd 280 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3939  𝒫 cpw 4541   class class class wbr 5062  dom cdm 5553  cres 5555  Fun wfun 6345  cfv 6351  Vtxcvtx 26697  iEdgciedg 26698  Edgcedg 26748   SubGraph csubgr 26965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-res 5565  df-iota 6311  df-fun 6353  df-fv 6359  df-subgr 26966
This theorem is referenced by:  uhgrspansubgr  26989
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