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Mirrors > Home > MPE Home > Th. List > subgrfun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
subgrfun | ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
2 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
4 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | eqid 2622 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
6 | 1, 2, 3, 4, 5 | subgrprop2 26166 | . . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))) |
7 | funss 5907 | . . . 4 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆))) | |
8 | 7 | 3ad2ant2 1083 | . . 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 446 | 1 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 Fun wfun 5882 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 SubGraph csubgr 26159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-subgr 26160 |
This theorem is referenced by: subgruhgrfun 26174 |
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