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Theorem issubgr2 26576
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 26575 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
763adant2 1165 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8 resss 5662 . . . . 5 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9 sseq1 3851 . . . . 5 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
108, 9mpbiri 250 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
11 funssres 6170 . . . . . . 7 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1211eqcomd 2831 . . . . . 6 ((Fun 𝐵𝐼𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
1312ex 403 . . . . 5 (Fun 𝐵 → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
14133ad2ant2 1168 . . . 4 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
1510, 14impbid2 218 . . 3 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼𝐵))
16153anbi2d 1569 . 2 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → ((𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
177, 16bitrd 271 1 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wss 3798  𝒫 cpw 4380   class class class wbr 4875  dom cdm 5346  cres 5348  Fun wfun 6121  cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352   SubGraph csubgr 26571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-res 5358  df-iota 6090  df-fun 6129  df-fv 6135  df-subgr 26572
This theorem is referenced by:  uhgrspansubgr  26595
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