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Theorem upgrspan 28539

Description: A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)

**Hypotheses**
Ref	Expression
uhgrspan.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspan.e	⊢ 𝐸 = (iEdg‘𝐺)
uhgrspan.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
uhgrspan.q	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
uhgrspan.r	⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
upgrspan.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)

**Assertion**
Ref	Expression
upgrspan	⊢ (𝜑 → 𝑆 ∈ UPGraph)

**Proof of Theorem upgrspan**
Step	Hyp	Ref	Expression
1		upgrspan.g	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
2		uhgrspan.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
3		uhgrspan.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
4		uhgrspan.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
5		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
6		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
7		upgruhgr 28351	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
9	2, 3, 4, 5, 6, 8	uhgrspansubgr 28537	. 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
10		subupgr 28533	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)
11	1, 9, 10	syl2anc 584	1 ⊢ (𝜑 → 𝑆 ∈ UPGraph)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ↾ cres 5677 ‘cfv 6540 Vtxcvtx 28245 iEdgciedg 28246 UHGraphcuhgr 28305 UPGraphcupgr 28329 SubGraph csubgr 28513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-edg 28297 df-uhgr 28307 df-upgr 28331 df-subgr 28514

This theorem is referenced by: upgrspanop 28543

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