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Theorem uspgrsprfo 42081
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 42079 . . 3 𝐹:𝐺𝑃
54a1i 11 . 2 (𝑉𝑊𝐹:𝐺𝑃)
61eleq2i 2722 . . . . . . 7 (𝑎𝑃𝑎 ∈ 𝒫 (Pairs‘𝑉))
7 selpw 4198 . . . . . . 7 (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
86, 7bitri 264 . . . . . 6 (𝑎𝑃𝑎 ⊆ (Pairs‘𝑉))
9 eqidd 2652 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑉 = 𝑉)
10 vex 3234 . . . . . . . . . . . . . . 15 𝑎 ∈ V
1110a1i 11 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ∈ V)
12 f1oi 6212 . . . . . . . . . . . . . . . . 17 ( I ↾ 𝑎):𝑎1-1-onto𝑎
1312a1i 11 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):𝑎1-1-onto𝑎)
14 dmresi 5492 . . . . . . . . . . . . . . . . 17 dom ( I ↾ 𝑎) = 𝑎
15 f1oeq2 6166 . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎)
18 sprvalpwle2 42064 . . . . . . . . . . . . . . . . 17 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
1918sseq2d 3666 . . . . . . . . . . . . . . . 16 (𝑉𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
2019biimpac 502 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
2117, 20jca 553 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
22 f1oeq3 6167 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎))
23 sseq1 3659 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
2422, 23anbi12d 747 . . . . . . . . . . . . . 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, 24elabd 3384 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
26 resiexg 7144 . . . . . . . . . . . . . . 15 (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
2710, 26ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ 𝑎) ∈ V
2827f11o 7170 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
3010a1i 11 . . . . . . . . . . . . . . . 16 (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V)
3130resiexd 6521 . . . . . . . . . . . . . . 15 (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
3231anim2i 592 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
3332ancoms 468 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
34 isuspgrop 26101 . . . . . . . . . . . . 13 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
3533, 34syl 17 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
3629, 35mpbird 247 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
37 fveq2 6229 . . . . . . . . . . . . . 14 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (Vtx‘𝑞) = (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩))
3837eqeq1d 2653 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
39 fveq2 6229 . . . . . . . . . . . . . 14 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (Edg‘𝑞) = (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩))
4039eqeq1d 2653 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4138, 40anbi12d 747 . . . . . . . . . . . 12 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
4241adantl 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)))
43 opvtxfv 25929 . . . . . . . . . . . . . 14 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
4432, 43syl 17 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
45 edgopval 25989 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
4632, 45syl 17 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
47 rnresi 5514 . . . . . . . . . . . . . 14 ran ( I ↾ 𝑎) = 𝑎
4846, 47syl6eq 2701 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
4944, 48jca 553 . . . . . . . . . . . 12 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
5049ancoms 468 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
5136, 42, 50rspcedvd 3348 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
529, 51jca 553 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
532eleq2i 2722 . . . . . . . . . 10 (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
5430anim1i 591 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑎 ∈ V ∧ 𝑉𝑊))
5554ancomd 466 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊𝑎 ∈ V))
56 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑣 = 𝑉 → (𝑣 = 𝑉𝑉 = 𝑉))
5756adantr 480 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (𝑣 = 𝑉𝑉 = 𝑉))
58 eqeq2 2662 . . . . . . . . . . . . . . 15 (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
59 eqeq2 2662 . . . . . . . . . . . . . . 15 (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
6058, 59bi2anan9 935 . . . . . . . . . . . . . 14 ((𝑣 = 𝑉𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
6160rexbidv 3081 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
6257, 61anbi12d 747 . . . . . . . . . . . 12 ((𝑣 = 𝑉𝑒 = 𝑎) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6362opelopabga 5017 . . . . . . . . . . 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))))
6455, 63syl 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))))
6553, 64syl5bb 272 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6652, 65mpbird 247 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
67 fveq2 6229 . . . . . . . . . 10 (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
6867eqeq2d 2661 . . . . . . . . 9 (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
6968adantl 481 . . . . . . . 8 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
70 op2ndg 7223 . . . . . . . . . . 11 ((𝑉𝑊𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7110, 70mpan2 707 . . . . . . . . . 10 (𝑉𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7271adantl 481 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7372eqcomd 2657 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
7466, 69, 73rspcedvd 3348 . . . . . . 7 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
7574ex 449 . . . . . 6 (𝑎 ⊆ (Pairs‘𝑉) → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
768, 75sylbi 207 . . . . 5 (𝑎𝑃 → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
7776impcom 445 . . . 4 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
781, 2, 3uspgrsprfv 42078 . . . . . . 7 (𝑏𝐺 → (𝐹𝑏) = (2nd𝑏))
7978adantl 481 . . . . . 6 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝐹𝑏) = (2nd𝑏))
8079eqeq2d 2661 . . . . 5 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (2nd𝑏)))
8180rexbidva 3078 . . . 4 ((𝑉𝑊𝑎𝑃) → (∃𝑏𝐺 𝑎 = (𝐹𝑏) ↔ ∃𝑏𝐺 𝑎 = (2nd𝑏)))
8277, 81mpbird 247 . . 3 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (𝐹𝑏))
8382ralrimiva 2995 . 2 (𝑉𝑊 → ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏))
84 dffo3 6414 . 2 (𝐹:𝐺onto𝑃 ↔ (𝐹:𝐺𝑃 ∧ ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏)))
855, 83, 84sylanbrc 699 1 (𝑉𝑊𝐹:𝐺onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cop 4216   class class class wbr 4685  {copab 4745  cmpt 4762   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  2nd c2nd 7209  cle 10113  2c2 11108  #chash 13157  Vtxcvtx 25919  Edgcedg 25984  USPGraphcuspgr 26088  Pairscspr 42052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-vtx 25921  df-iedg 25922  df-edg 25985  df-upgr 26022  df-uspgr 26090  df-spr 42053
This theorem is referenced by:  uspgrsprf1o  42082
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