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Theorem upgrspan 29440

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 29249	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
9	2, 3, 4, 5, 6, 8	uhgrspansubgr 29438	. 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)
10		subupgr 29434	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)
11	1, 9, 10	syl2anc 593	1 ⊢ (𝜑 → 𝑆 ∈ UPGraph)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 class class class wbr 5099 ↾ cres 5647 ‘cfv 6517 Vtxcvtx 29143 iEdgciedg 29144 UHGraphcuhgr 29203 UPGraphcupgr 29227 SubGraph csubgr 29414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fv 6525 df-edg 29195 df-uhgr 29205 df-upgr 29229 df-subgr 29415

This theorem is referenced by: upgrspanop 29444

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