opvtxfv - Intuitionistic Logic Explorer

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Theorem opvtxfv 16267

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 4824	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 16266	. . 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 6384	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2271	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 Vcvv 2821 ⟨cop 3712 × cxp 4772 ‘cfv 5377 1^st c1st 6372 Vtxcvtx 16257

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-cnex 8270 ax-resscn 8271 ax-1re 8273 ax-addrcl 8276

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-fo 5383 df-fv 5385 df-1st 6374 df-inn 9305 df-ndx 13355 df-slot 13356 df-base 13358 df-vtx 16259

This theorem is used by: opvtxov 16268 opvtxfvi 16272 gropd 16292 isuhgropm 16326 uhgrunop 16332 upgrop 16349 upgr1eopdc 16368 upgr1een 16369 umgr1een 16370 upgrunop 16372 umgrunop 16374 isuspgropen 16409 isusgropen 16410 ausgrusgrben 16413 uspgr1eopdc 16488 usgr1eop 16490 uhgrspanop 16527 vtxdgop 16537 p1evtxdeqfilem 16556 p1evtxdeqfi 16557 p1evtxdp1fi 16558 eupthvdres 16720 eupth2lem3fi 16721 eupth2lembfi 16722 konigsbergvtx 16727 konigsberg 16738

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