opvtxfv - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > opvtxfv		GIF version

Theorem opvtxfv 16468

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 4826	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 16467	. . 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 6388	. 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 4774 ‘cfv 5380 1^st c1st 6376 Vtxcvtx 16458

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 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-cnex 8276 ax-resscn 8277 ax-1re 8279 ax-addrcl 8282

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 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-fo 5386 df-fv 5388 df-1st 6378 df-inn 9314 df-ndx 13413 df-slot 13414 df-base 13416 df-vtx 16460

This theorem is used by: opvtxov 16469 opvtxfvi 16473 gropd 16493 isuhgropm 16527 uhgrunop 16533 upgrop 16550 upgr1eopdc 16569 upgr1een 16570 umgr1een 16571 upgrunop 16573 umgrunop 16575 isuspgropen 16610 isusgropen 16611 ausgrusgrben 16614 uspgr1eopdc 16689 usgr1eop 16691 uhgrspanop 16728 vtxdgop 16738 p1evtxdeqfilem 16757 p1evtxdeqfi 16758 p1evtxdp1fi 16759 eupthvdres 16921 eupth2lem3fi 16922 eupth2lembfi 16923 konigsbergvtx 16928 konigsberg 16939

W3C validator