Theorem subgrfun 29354

Description: The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)

Assertion

Ref	Expression
subgrfun	⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Proof of Theorem subgrfun

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 29347	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		funss 6511	. . . 4 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
8	7	3ad2ant2 1134	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
9	6, 8	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
10	9	impcom 407	1 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098  Fun wfun 6486  ‘cfv 6492  Vtxcvtx 29069  iEdgciedg 29070  Edgcedg 29120   SubGraph csubgr 29340

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-subgr 29341

This theorem is referenced by:  subgruhgrfun  29355