opvtxfv - Intuitionistic Logic Explorer

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Theorem opvtxfv 16143

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)

**Assertion**
Ref	Expression
opvtxfv	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

**Proof of Theorem opvtxfv**
Step	Hyp	Ref	Expression
1		opelvvg 4804	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 16142	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 14	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
4		op1stg 6357	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2267	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2205 Vcvv 2815 ⟨cop 3697 × cxp 4752 ‘cfv 5357 1^st c1st 6345 Vtxcvtx 16133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-cnex 8234 ax-resscn 8235 ax-1re 8237 ax-addrcl 8240

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-un 3218 df-in 3220 df-ss 3227 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fo 5363 df-fv 5365 df-1st 6347 df-inn 9255 df-ndx 13299 df-slot 13300 df-base 13302 df-vtx 16135

This theorem is referenced by: opvtxov 16144 opvtxfvi 16148 gropd 16168 isuhgropm 16202 uhgrunop 16208 upgrop 16225 upgr1eopdc 16244 upgr1een 16245 umgr1een 16246 upgrunop 16248 umgrunop 16250 isuspgropen 16285 isusgropen 16286 ausgrusgrben 16289 uspgr1eopdc 16364 usgr1eop 16366 uhgrspanop 16403 vtxdgop 16413 p1evtxdeqfilem 16432 p1evtxdeqfi 16433 p1evtxdp1fi 16434 eupthvdres 16596 eupth2lem3fi 16597 eupth2lembfi 16598 konigsbergvtx 16603 konigsberg 16614

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