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Theorem issubgr2 26057
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 26056 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
763adant2 1078 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8 resss 5381 . . . . 5 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9 sseq1 3605 . . . . 5 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
108, 9mpbiri 248 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
11 funssres 5888 . . . . . . 7 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1211eqcomd 2627 . . . . . 6 ((Fun 𝐵𝐼𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
1312ex 450 . . . . 5 (Fun 𝐵 → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
14133ad2ant2 1081 . . . 4 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
1510, 14impbid2 216 . . 3 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼𝐵))
16153anbi2d 1401 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → ((𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
177, 16bitrd 268 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3555  𝒫 cpw 4130   class class class wbr 4613  dom cdm 5074  cres 5076  Fun wfun 5841  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   SubGraph csubgr 26052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-subgr 26053
This theorem is referenced by:  uhgrspansubgr  26076
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