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Theorem subgreldmiedg 29301

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 2736	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2736	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2736	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29292	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		dmss 5912	. . . . 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 3981	. . 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 SubGraph csubgr 29285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 df-res 5696 df-iota 6513 df-fv 6568 df-subgr 29286

This theorem is referenced by: subgruhgredgd 29302 subumgredg2 29303 subupgr 29305

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