Theorem subgreldmiedg 29256

Description: An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)

Assertion

Ref	Expression
subgreldmiedg	⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))

Proof of Theorem subgreldmiedg

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2731	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2731	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2731	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2731	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29247	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		dmss 5837	. . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
8	7	3ad2ant2 1134	. . . 4 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
9	8	sseld 3928	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺)))
10	6, 9	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺)))
11	10	imp 406	1 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4545   class class class wbr 5086  dom cdm 5611  ‘cfv 6476  Vtxcvtx 28969  iEdgciedg 28970  Edgcedg 29020   SubGraph csubgr 29240

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-dm 5621  df-res 5623  df-iota 6432  df-fv 6484  df-subgr 29241

This theorem is referenced by:  subgruhgredgd  29257  subumgredg2  29258  subupgr  29260