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Theorem upgrspan 29552

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 29361	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1, 7	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
9	2, 3, 4, 5, 6, 8	uhgrspansubgr 29550	. 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
10		subupgr 29546	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)
11	1, 9, 10	syl2anc 595	1 ⊢ (𝜑 → 𝑆 ∈ UPGraph)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1563 ∈ wcel 2145 class class class wbr 5105 ↾ cres 5654 ‘cfv 6525 Vtxcvtx 29255 iEdgciedg 29256 UHGraphcuhgr 29315 UPGraphcupgr 29339 SubGraph csubgr 29526

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-fv 6533 df-edg 29307 df-uhgr 29317 df-upgr 29341 df-subgr 29527

This theorem is referenced by: upgrspanop 29556

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