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Mirrors > Home > MPE Home > Th. List > edgaval | Structured version Visualization version GIF version |
Description: The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
edgaval | ⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-edga 25793 | . . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))) |
3 | fveq2 6103 | . . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) | |
4 | 3 | rneqd 5274 | . . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺)) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺)) |
6 | elex 3185 | . 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
7 | fvex 6113 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
8 | 7 | rnex 6992 | . . 3 ⊢ ran (iEdg‘𝐺) ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑉 → ran (iEdg‘𝐺) ∈ V) |
10 | 2, 5, 6, 9 | fvmptd 6197 | 1 ⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 iEdgciedg 25674 Edgcedga 25792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-edga 25793 |
This theorem is referenced by: edgaopval 25795 edgastruct 25797 edgiedgb 25798 edg0iedg0 25800 uhgredgn0 25802 upgredgss 25806 umgredgss 25807 edgupgr 25808 uhgrvtxedgiedgb 25810 upgredg 25811 usgredgss 40390 ausgrumgri 40397 ausgrusgri 40398 uspgrf1oedg 40403 uspgrupgrushgr 40407 usgrumgruspgr 40410 usgruspgrb 40411 usgrf1oedg 40434 uhgr2edg 40435 usgrsizedg 40442 usgredg3 40443 ushgredgedga 40456 ushgredgedgaloop 40458 usgr1e 40471 edg0usgr 40479 usgr1v0edg 40483 usgrexmpledg 40486 subgrprop3 40500 0grsubgr 40502 0uhgrsubgr 40503 subgruhgredgd 40508 uhgrspansubgrlem 40514 uhgrspan1 40527 upgrres1 40532 usgredgffibi 40543 dfnbgr3 40562 nbupgrres 40592 usgrnbcnvfv 40593 cplgrop 40659 cusgrexi 40662 cusgrsize 40670 1loopgredg 40716 1egrvtxdg0 40727 umgr2v2eedg 40740 edginwlk 40839 1wlkl1loop 40842 1wlkvtxedg 40852 uspgr2wlkeq 40854 1wlkiswwlks1 41064 1wlkiswwlks2lem4 41069 1wlkiswwlks2lem5 41070 1wlkiswwlks2 41072 1wlkiswwlksupgr2 41074 2pthon3v-av 41150 umgrwwlks2on 41161 clwlkclwwlk 41211 clwlksfclwwlk 41269 |
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