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Theorem subupgr 26089
Description: A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
Assertion
Ref Expression
subupgr ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )

Proof of Theorem subupgr
Dummy variables 𝑥 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2621 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2621 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 26076 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 upgruhgr 25909 . . . . . . . . . 10 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
8 subgruhgrfun 26084 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
97, 8sylan 488 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
109ancoms 469 . . . . . . . 8 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → Fun (iEdg‘𝑆))
11 funfn 5882 . . . . . . . 8 (Fun (iEdg‘𝑆) ↔ (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1210, 11sylib 208 . . . . . . 7 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1312adantl 482 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
147anim2i 592 . . . . . . . . . . . . . 14 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ))
1514adantl 482 . . . . . . . . . . . . 13 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph ))
1615ancomd 467 . . . . . . . . . . . 12 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺))
1716anim1i 591 . . . . . . . . . . 11 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)))
1817simplld 790 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph )
19 simpl 473 . . . . . . . . . . . 12 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → 𝑆 SubGraph 𝐺)
2019adantl 482 . . . . . . . . . . 11 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → 𝑆 SubGraph 𝐺)
2120adantr 481 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
22 simpr 477 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
231, 3, 18, 21, 22subgruhgredgd 26086 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
244uhgrfun 25874 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
257, 24syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
2625ad2antll 764 . . . . . . . . . . . . . 14 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → Fun (iEdg‘𝐺))
2726adantr 481 . . . . . . . . . . . . 13 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → Fun (iEdg‘𝐺))
28 simpll2 1099 . . . . . . . . . . . . 13 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
29 funssfv 6171 . . . . . . . . . . . . 13 ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
3027, 28, 22, 29syl3anc 1323 . . . . . . . . . . . 12 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
3130eqcomd 2627 . . . . . . . . . . 11 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
3231fveq2d 6157 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (#‘((iEdg‘𝑆)‘𝑥)) = (#‘((iEdg‘𝐺)‘𝑥)))
33 subgreldmiedg 26085 . . . . . . . . . . . . . . 15 ((𝑆 SubGraph 𝐺𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝐺))
3433ex 450 . . . . . . . . . . . . . 14 (𝑆 SubGraph 𝐺 → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
3534adantr 481 . . . . . . . . . . . . 13 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
3635adantl 482 . . . . . . . . . . . 12 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
37 simpr 477 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → 𝐺 ∈ UPGraph )
38 funfn 5882 . . . . . . . . . . . . . . . . 17 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
3925, 38sylib 208 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
4039adantl 482 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
41 simpl 473 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → 𝑥 ∈ dom (iEdg‘𝐺))
422, 4upgrle 25898 . . . . . . . . . . . . . . 15 ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
4337, 40, 41, 42syl3anc 1323 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
4443expcom 451 . . . . . . . . . . . . 13 (𝐺 ∈ UPGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
4544ad2antll 764 . . . . . . . . . . . 12 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝑥 ∈ dom (iEdg‘𝐺) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
4636, 45syld 47 . . . . . . . . . . 11 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝑥 ∈ dom (iEdg‘𝑆) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
4746imp 445 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (#‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
4832, 47eqbrtrd 4640 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (#‘((iEdg‘𝑆)‘𝑥)) ≤ 2)
49 fveq2 6153 . . . . . . . . . . 11 (𝑒 = ((iEdg‘𝑆)‘𝑥) → (#‘𝑒) = (#‘((iEdg‘𝑆)‘𝑥)))
5049breq1d 4628 . . . . . . . . . 10 (𝑒 = ((iEdg‘𝑆)‘𝑥) → ((#‘𝑒) ≤ 2 ↔ (#‘((iEdg‘𝑆)‘𝑥)) ≤ 2))
5150elrab 3350 . . . . . . . . 9 (((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2} ↔ (((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∧ (#‘((iEdg‘𝑆)‘𝑥)) ≤ 2))
5223, 48, 51sylanbrc 697 . . . . . . . 8 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2})
5352ralrimiva 2961 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2})
54 fnfvrnss 6351 . . . . . . 7 (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2})
5513, 53, 54syl2anc 692 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2})
56 df-f 5856 . . . . . 6 ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2} ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
5713, 55, 56sylanbrc 697 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2})
58 subgrv 26072 . . . . . . . 8 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
591, 3isupgr 25892 . . . . . . . . 9 (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
6059adantr 481 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
6158, 60syl 17 . . . . . . 7 (𝑆 SubGraph 𝐺 → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
6261adantr 481 . . . . . 6 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
6362adantl 482 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (#‘𝑒) ≤ 2}))
6457, 63mpbird 247 . . . 4 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph )) → 𝑆 ∈ UPGraph )
6564ex 450 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → 𝑆 ∈ UPGraph ))
666, 65syl 17 . 2 (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph ) → 𝑆 ∈ UPGraph ))
6766anabsi8 860 1 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cdif 3556  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  dom cdm 5079  ran crn 5080  Fun wfun 5846   Fn wfn 5847  wf 5848  cfv 5852  cle 10027  2c2 11022  #chash 13065  Vtxcvtx 25791  iEdgciedg 25792  Edgcedg 25856   UHGraph cuhgr 25864   UPGraph cupgr 25888   SubGraph csubgr 26069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-edg 25857  df-uhgr 25866  df-upgr 25890  df-subgr 26070
This theorem is referenced by:  upgrspan  26095
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