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Theorem upgrspan 26185

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 25997	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph )
9	2, 3, 4, 5, 6, 8	uhgrspansubgr 26183	. 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
10		subupgr 26179	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )
11	1, 9, 10	syl2anc 693	1 ⊢ (𝜑 → 𝑆 ∈ UPGraph )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ↾ cres 5116 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 UPGraph cupgr 25975 SubGraph csubgr 26159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-subgr 26160

This theorem is referenced by: upgrspanop 26189

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