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Theorem upgrspan 29227

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 29036	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
9	2, 3, 4, 5, 6, 8	uhgrspansubgr 29225	. 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
10		subupgr 29221	. 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 1540 ∈ wcel 2109 class class class wbr 5110 ↾ cres 5643 ‘cfv 6514 Vtxcvtx 28930 iEdgciedg 28931 UHGraphcuhgr 28990 UPGraphcupgr 29014 SubGraph csubgr 29201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-edg 28982 df-uhgr 28992 df-upgr 29016 df-subgr 29202

This theorem is referenced by: upgrspanop 29231

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