Theorem issubgr2 27056

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	⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))

Proof of Theorem issubgr2

Step	Hyp	Ref	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)
6	1, 2, 3, 4, 5	issubgr 27055	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
7	6	3adant2 1127	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8		resss 5880	. . . . 5 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
9		sseq1 3994	. . . . 5 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
10	8, 9	mpbiri 260	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
11		funssres 6400	. . . . . . 7 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼)
12	11	eqcomd 2829	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼))
13	12	ex 415	. . . . 5 ⊢ (Fun 𝐵 → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
14	13	3ad2ant2 1130	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼)))
15	10, 14	impbid2 228	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼 ⊆ 𝐵))
16	15	3anbi2d 1437	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))
17	7, 16	bitrd 281	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  𝒫 cpw 4541   class class class wbr 5068  dom cdm 5557   ↾ cres 5559  Fun wfun 6351  ‘cfv 6357  Vtxcvtx 26783  iEdgciedg 26784  Edgcedg 26834   SubGraph csubgr 27051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-subgr 27052

This theorem is referenced by:  uhgrspansubgr  27075