MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubgr2 Structured version   Visualization version   GIF version

Theorem issubgr2 26387
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 26386 . . 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 5562 . . . . 5 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9 sseq1 3775 . . . . 5 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
108, 9mpbiri 248 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
11 funssres 6072 . . . . . . 7 ((Fun 𝐵𝐼𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
1211eqcomd 2777 . . . . . 6 ((Fun 𝐵𝐼𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
1312ex 397 . . . . 5 (Fun 𝐵 → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
14133ad2ant2 1128 . . . 4 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼𝐵𝐼 = (𝐵 ↾ dom 𝐼)))
1510, 14impbid2 216 . . 3 ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼𝐵))
16153anbi2d 1552 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wss 3723  𝒫 cpw 4298   class class class wbr 4787  dom cdm 5250  cres 5252  Fun wfun 6024  cfv 6030  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160   SubGraph csubgr 26382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-subgr 26383
This theorem is referenced by:  uhgrspansubgr  26406
  Copyright terms: Public domain W3C validator