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Theorem subupgr 29319
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 2735 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2735 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2735 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 29306 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 upgruhgr 29134 . . . . . . . . . 10 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8 subgruhgrfun 29314 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
97, 8sylan 580 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
109ancoms 458 . . . . . . . 8 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → Fun (iEdg‘𝑆))
1110funfnd 6599 . . . . . . 7 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1211adantl 481 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
13 fveq2 6907 . . . . . . . . . 10 (𝑒 = ((iEdg‘𝑆)‘𝑥) → (♯‘𝑒) = (♯‘((iEdg‘𝑆)‘𝑥)))
1413breq1d 5158 . . . . . . . . 9 (𝑒 = ((iEdg‘𝑆)‘𝑥) → ((♯‘𝑒) ≤ 2 ↔ (♯‘((iEdg‘𝑆)‘𝑥)) ≤ 2))
157anim2i 617 . . . . . . . . . . . . . 14 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph))
1615adantl 481 . . . . . . . . . . . . 13 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → (𝑆 SubGraph 𝐺𝐺 ∈ UHGraph))
1716ancomd 461 . . . . . . . . . . . 12 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → (𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺))
1817anim1i 615 . . . . . . . . . . 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)))
1918simplld 768 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph)
20 simpl 482 . . . . . . . . . . . 12 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → 𝑆 SubGraph 𝐺)
2120adantl 481 . . . . . . . . . . 11 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → 𝑆 SubGraph 𝐺)
2221adantr 480 . . . . . . . . . 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)
23 simpr 484 . . . . . . . . . 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))
241, 3, 19, 22, 23subgruhgredgd 29316 . . . . . . . . 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) ∖ {∅}))
254uhgrfun 29098 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
267, 25syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
2726ad2antll 729 . . . . . . . . . . . . . 14 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → Fun (iEdg‘𝐺))
2827adantr 480 . . . . . . . . . . . . 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))
29 simpll2 1212 . . . . . . . . . . . . 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))
30 funssfv 6928 . . . . . . . . . . . . 13 ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
3128, 29, 23, 30syl3anc 1370 . . . . . . . . . . . 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))
3231eqcomd 2741 . . . . . . . . . . 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))
3332fveq2d 6911 . . . . . . . . . 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)))
34 subgreldmiedg 29315 . . . . . . . . . . . . . . 15 ((𝑆 SubGraph 𝐺𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝐺))
3534ex 412 . . . . . . . . . . . . . 14 (𝑆 SubGraph 𝐺 → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
3635adantr 480 . . . . . . . . . . . . 13 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
3736adantl 481 . . . . . . . . . . . 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)))
38 simpr 484 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
3926funfnd 6599 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
4039adantl 481 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
41 simpl 482 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝑥 ∈ dom (iEdg‘𝐺))
422, 4upgrle 29122 . . . . . . . . . . . . . . 15 ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
4338, 40, 41, 42syl3anc 1370 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
4443expcom 413 . . . . . . . . . . . . 13 (𝐺 ∈ UPGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
4544ad2antll 729 . . . . . . . . . . . 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))
4637, 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 406 . . . . . . . . . 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)
4833, 47eqbrtrd 5170 . . . . . . . . 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)
4914, 24, 48elrabd 3697 . . . . . . . 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})
5049ralrimiva 3144 . . . . . . 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})
51 fnfvrnss 7141 . . . . . . 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})
5212, 50, 51syl2anc 584 . . . . . 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})
53 df-f 6567 . . . . . 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}))
5412, 52, 53sylanbrc 583 . . . . 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})
55 subgrv 29302 . . . . . . . 8 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
561, 3isupgr 29116 . . . . . . . . 9 (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
5756adantr 480 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
5855, 57syl 17 . . . . . . 7 (𝑆 SubGraph 𝐺 → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
5958adantr 480 . . . . . 6 ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
6059adantl 481 . . . . 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}))
6154, 60mpbird 257 . . . 4 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UPGraph)) → 𝑆 ∈ UPGraph)
6261ex 412 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
636, 62syl 17 . 2 (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
6463anabsi8 672 1 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  UHGraphcuhgr 29088  UPGraphcupgr 29112   SubGraph csubgr 29299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-subgr 29300
This theorem is referenced by:  upgrspan  29325  isubgrupgr  47794
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