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Theorem uspgrsprfo 47992
Description: The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
Hypotheses
Ref Expression
uspgrsprf.p 𝑃 = 𝒫 (Pairs‘𝑉)
uspgrsprf.g 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
uspgrsprf.f 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
Assertion
Ref Expression
uspgrsprfo (𝑉𝑊𝐹:𝐺onto𝑃)
Distinct variable groups:   𝑃,𝑒,𝑞,𝑣   𝑒,𝑉,𝑞,𝑣   𝑒,𝑊,𝑣   𝑔,𝐺   𝑃,𝑔,𝑒,𝑣   𝑊,𝑞
Allowed substitution hints:   𝐹(𝑣,𝑒,𝑔,𝑞)   𝐺(𝑣,𝑒,𝑞)   𝑉(𝑔)   𝑊(𝑔)

Proof of Theorem uspgrsprfo
Dummy variables 𝑎 𝑏 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrsprf.p . . . 4 𝑃 = 𝒫 (Pairs‘𝑉)
2 uspgrsprf.g . . . 4 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
3 uspgrsprf.f . . . 4 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
41, 2, 3uspgrsprf 47990 . . 3 𝐹:𝐺𝑃
54a1i 11 . 2 (𝑉𝑊𝐹:𝐺𝑃)
61eleq2i 2831 . . . . . . 7 (𝑎𝑃𝑎 ∈ 𝒫 (Pairs‘𝑉))
7 velpw 4610 . . . . . . 7 (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
86, 7bitri 275 . . . . . 6 (𝑎𝑃𝑎 ⊆ (Pairs‘𝑉))
9 eqidd 2736 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑉 = 𝑉)
10 vex 3482 . . . . . . . . . . . . . . 15 𝑎 ∈ V
1110a1i 11 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ∈ V)
12 f1oi 6887 . . . . . . . . . . . . . . . . 17 ( I ↾ 𝑎):𝑎1-1-onto𝑎
1312a1i 11 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):𝑎1-1-onto𝑎)
14 dmresi 6072 . . . . . . . . . . . . . . . . 17 dom ( I ↾ 𝑎) = 𝑎
15 f1oeq2 6838 . . . . . . . . . . . . . . . . 17 (dom ( I ↾ 𝑎) = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎 ↔ ( I ↾ 𝑎):𝑎1-1-onto𝑎))
1614, 15ax-mp 5 . . . . . . . . . . . . . . . 16 (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎 ↔ ( I ↾ 𝑎):𝑎1-1-onto𝑎)
1713, 16sylibr 234 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎)
18 sprvalpwle2 47414 . . . . . . . . . . . . . . . . 17 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
1918sseq2d 4028 . . . . . . . . . . . . . . . 16 (𝑉𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2019biimpac 478 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
2117, 20jca 511 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22 f1oeq3 6839 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎))
23 sseq1 4021 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2422, 23anbi12d 632 . . . . . . . . . . . . . 14 (𝑓 = 𝑎 → ((( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) ↔ (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
2511, 21, 24spcedv 3598 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
26 resiexg 7935 . . . . . . . . . . . . . . 15 (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
2710, 26ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ 𝑎) ∈ V
2827f11o 7970 . . . . . . . . . . . . 13 (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2925, 28sylibr 234 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
3010a1i 11 . . . . . . . . . . . . . . 15 (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V)
3130resiexd 7236 . . . . . . . . . . . . . 14 (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
3231anim1ci 616 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
33 isuspgrop 29193 . . . . . . . . . . . . 13 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
3432, 33syl 17 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
3529, 34mpbird 257 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
36 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
37 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
3836, 37anbi12d 632 . . . . . . . . . . . 12 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
3938adantl 481 . . . . . . . . . . 11 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) ∧ 𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩) → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
40 opvtxfv 29036 . . . . . . . . . . . . . 14 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
4131, 40sylan2 593 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
42 edgopval 29083 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
4331, 42sylan2 593 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
44 rnresi 6095 . . . . . . . . . . . . . 14 ran ( I ↾ 𝑎) = 𝑎
4543, 44eqtrdi 2791 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
4641, 45jca 511 . . . . . . . . . . . 12 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4746ancoms 458 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4835, 39, 47rspcedvd 3624 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
499, 48jca 511 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
502eleq2i 2831 . . . . . . . . . 10 (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
5130anim1ci 616 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊𝑎 ∈ V))
52 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑣 = 𝑉 → (𝑣 = 𝑉𝑉 = 𝑉))
5352adantr 480 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (𝑣 = 𝑉𝑉 = 𝑉))
54 eqeq2 2747 . . . . . . . . . . . . . . 15 (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
55 eqeq2 2747 . . . . . . . . . . . . . . 15 (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
5654, 55bi2anan9 638 . . . . . . . . . . . . . 14 ((𝑣 = 𝑉𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
5756rexbidv 3177 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
5853, 57anbi12d 632 . . . . . . . . . . . 12 ((𝑣 = 𝑉𝑒 = 𝑎) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
5958opelopabga 5543 . . . . . . . . . . 11 ((𝑉𝑊𝑎 ∈ V) → (⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6051, 59syl 17 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6150, 60bitrid 283 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6249, 61mpbird 257 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
63 fveq2 6907 . . . . . . . . . 10 (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
6463eqeq2d 2746 . . . . . . . . 9 (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
6564adantl 481 . . . . . . . 8 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
66 op2ndg 8026 . . . . . . . . . . 11 ((𝑉𝑊𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
6766elvd 3484 . . . . . . . . . 10 (𝑉𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
6867adantl 481 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
6968eqcomd 2741 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
7062, 65, 69rspcedvd 3624 . . . . . . 7 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
7170ex 412 . . . . . 6 (𝑎 ⊆ (Pairs‘𝑉) → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
728, 71sylbi 217 . . . . 5 (𝑎𝑃 → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
7372impcom 407 . . . 4 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
741, 2, 3uspgrsprfv 47989 . . . . . . 7 (𝑏𝐺 → (𝐹𝑏) = (2nd𝑏))
7574adantl 481 . . . . . 6 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝐹𝑏) = (2nd𝑏))
7675eqeq2d 2746 . . . . 5 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (2nd𝑏)))
7776rexbidva 3175 . . . 4 ((𝑉𝑊𝑎𝑃) → (∃𝑏𝐺 𝑎 = (𝐹𝑏) ↔ ∃𝑏𝐺 𝑎 = (2nd𝑏)))
7873, 77mpbird 257 . . 3 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (𝐹𝑏))
7978ralrimiva 3144 . 2 (𝑉𝑊 → ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏))
80 dffo3 7122 . 2 (𝐹:𝐺onto𝑃 ↔ (𝐹:𝐺𝑃 ∧ ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏)))
815, 79, 80sylanbrc 583 1 (𝑉𝑊𝐹:𝐺onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  cop 4637   class class class wbr 5148  {copab 5210  cmpt 5231   I cid 5582  dom cdm 5689  ran crn 5690  cres 5691  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  2nd c2nd 8012  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  Edgcedg 29079  USPGraphcuspgr 29180  Pairscspr 47402
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-vtx 29030  df-iedg 29031  df-edg 29080  df-upgr 29114  df-uspgr 29182  df-spr 47403
This theorem is referenced by:  uspgrsprf1o  47993
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