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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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