opvtxfv - Intuitionistic Logic Explorer

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Theorem opvtxfv 15946

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 4781	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 15945	. . 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 6322	. 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 1398 ∈ wcel 2202 Vcvv 2803 ⟨cop 3676 × cxp 4729 ‘cfv 5333 1^st c1st 6310 Vtxcvtx 15936

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-cnex 8166 ax-resscn 8167 ax-1re 8169 ax-addrcl 8172

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-un 3205 df-in 3207 df-ss 3214 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fo 5339 df-fv 5341 df-1st 6312 df-inn 9186 df-ndx 13148 df-slot 13149 df-base 13151 df-vtx 15938

This theorem is referenced by: opvtxov 15947 opvtxfvi 15951 gropd 15971 isuhgropm 16005 uhgrunop 16011 upgrop 16028 upgr1eopdc 16047 upgr1een 16048 umgr1een 16049 upgrunop 16051 umgrunop 16053 isuspgropen 16088 isusgropen 16089 ausgrusgrben 16092 uspgr1eopdc 16167 usgr1eop 16169 uhgrspanop 16206 vtxdgop 16216 p1evtxdeqfilem 16235 p1evtxdeqfi 16236 p1evtxdp1fi 16237 eupthvdres 16399 eupth2lem3fi 16400 eupth2lembfi 16401 konigsbergvtx 16406 konigsberg 16417

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