opvtxfv - Intuitionistic Logic Explorer

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Theorem opvtxfv 15879

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 4775	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 15878	. . 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 6313	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2264	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1397 ∈ wcel 2202 Vcvv 2802 ⟨cop 3672 × cxp 4723 ‘cfv 5326 1^st c1st 6301 Vtxcvtx 15869

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-cnex 8123 ax-resscn 8124 ax-1re 8126 ax-addrcl 8129

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-un 3204 df-in 3206 df-ss 3213 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fo 5332 df-fv 5334 df-1st 6303 df-inn 9144 df-ndx 13090 df-slot 13091 df-base 13093 df-vtx 15871

This theorem is referenced by: opvtxov 15880 opvtxfvi 15884 gropd 15904 isuhgropm 15938 uhgrunop 15944 upgrop 15961 upgr1eopdc 15980 upgr1een 15981 umgr1een 15982 upgrunop 15984 umgrunop 15986 isuspgropen 16021 isusgropen 16022 ausgrusgrben 16025 uspgr1eopdc 16100 usgr1eop 16102 uhgrspanop 16139 vtxdgop 16149 p1evtxdeqfilem 16168 p1evtxdeqfi 16169 p1evtxdp1fi 16170 eupthvdres 16332 eupth2lem3fi 16333 eupth2lembfi 16334

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