Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uspgrsprf1 Structured version   Visualization version   GIF version

Theorem uspgrsprf1 42080
 Description: The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into 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
uspgrsprf1 𝐹:𝐺1-1𝑃
Distinct variable groups:   𝑃,𝑒,𝑞,𝑣   𝑒,𝑉,𝑞,𝑣   𝑔,𝐺   𝑃,𝑔,𝑒,𝑣
Allowed substitution hints:   𝐹(𝑣,𝑒,𝑔,𝑞)   𝐺(𝑣,𝑒,𝑞)   𝑉(𝑔)

Proof of Theorem uspgrsprf1
Dummy variables 𝑎 𝑏 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrsprf.p . . 3 𝑃 = 𝒫 (Pairs‘𝑉)
2 uspgrsprf.g . . 3 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
3 uspgrsprf.f . . 3 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
41, 2, 3uspgrsprf 42079 . 2 𝐹:𝐺𝑃
51, 2, 3uspgrsprfv 42078 . . . . 5 (𝑎𝐺 → (𝐹𝑎) = (2nd𝑎))
61, 2, 3uspgrsprfv 42078 . . . . 5 (𝑏𝐺 → (𝐹𝑏) = (2nd𝑏))
75, 6eqeqan12d 2667 . . . 4 ((𝑎𝐺𝑏𝐺) → ((𝐹𝑎) = (𝐹𝑏) ↔ (2nd𝑎) = (2nd𝑏)))
82eleq2i 2722 . . . . . 6 (𝑎𝐺𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
9 elopab 5012 . . . . . 6 (𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣𝑒(𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
10 opeq12 4435 . . . . . . . . 9 ((𝑣 = 𝑤𝑒 = 𝑓) → ⟨𝑣, 𝑒⟩ = ⟨𝑤, 𝑓⟩)
1110eqeq2d 2661 . . . . . . . 8 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑎 = ⟨𝑣, 𝑒⟩ ↔ 𝑎 = ⟨𝑤, 𝑓⟩))
12 eqeq1 2655 . . . . . . . . . 10 (𝑣 = 𝑤 → (𝑣 = 𝑉𝑤 = 𝑉))
1312adantr 480 . . . . . . . . 9 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑣 = 𝑉𝑤 = 𝑉))
14 eqeq2 2662 . . . . . . . . . . 11 (𝑣 = 𝑤 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑤))
15 eqeq2 2662 . . . . . . . . . . 11 (𝑒 = 𝑓 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑓))
1614, 15bi2anan9 935 . . . . . . . . . 10 ((𝑣 = 𝑤𝑒 = 𝑓) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
1716rexbidv 3081 . . . . . . . . 9 ((𝑣 = 𝑤𝑒 = 𝑓) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
1813, 17anbi12d 747 . . . . . . . 8 ((𝑣 = 𝑤𝑒 = 𝑓) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
1911, 18anbi12d 747 . . . . . . 7 ((𝑣 = 𝑤𝑒 = 𝑓) → ((𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ↔ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))))
2019cbvex2v 2323 . . . . . 6 (∃𝑣𝑒(𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ↔ ∃𝑤𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
218, 9, 203bitri 286 . . . . 5 (𝑎𝐺 ↔ ∃𝑤𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
222eleq2i 2722 . . . . . 6 (𝑏𝐺𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
23 elopab 5012 . . . . . 6 (𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
2422, 23bitri 264 . . . . 5 (𝑏𝐺 ↔ ∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
25 eqeq2 2662 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑉 → (𝑣 = 𝑤𝑣 = 𝑉))
26 opeq12 4435 . . . . . . . . . . . . . . . . . 18 ((𝑤 = 𝑣𝑓 = 𝑒) → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)
2726ex 449 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑣 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
2827equcoms 1993 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
2925, 28syl6bir 244 . . . . . . . . . . . . . . 15 (𝑤 = 𝑉 → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
3029ad2antrl 764 . . . . . . . . . . . . . 14 ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
3130com12 32 . . . . . . . . . . . . 13 (𝑣 = 𝑉 → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
3231ad2antrl 764 . . . . . . . . . . . 12 ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
3332imp 444 . . . . . . . . . . 11 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
34 vex 3234 . . . . . . . . . . . . . 14 𝑤 ∈ V
35 vex 3234 . . . . . . . . . . . . . 14 𝑓 ∈ V
3634, 35op2ndd 7221 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑤, 𝑓⟩ → (2nd𝑎) = 𝑓)
3736ad2antrl 764 . . . . . . . . . . . 12 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (2nd𝑎) = 𝑓)
38 vex 3234 . . . . . . . . . . . . . . 15 𝑣 ∈ V
39 vex 3234 . . . . . . . . . . . . . . 15 𝑒 ∈ V
4038, 39op2ndd 7221 . . . . . . . . . . . . . 14 (𝑏 = ⟨𝑣, 𝑒⟩ → (2nd𝑏) = 𝑒)
4140adantr 480 . . . . . . . . . . . . 13 ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑏) = 𝑒)
4241adantr 480 . . . . . . . . . . . 12 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (2nd𝑏) = 𝑒)
4337, 42eqeq12d 2666 . . . . . . . . . . 11 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → ((2nd𝑎) = (2nd𝑏) ↔ 𝑓 = 𝑒))
44 eqeq12 2664 . . . . . . . . . . . . . . . 16 ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ 𝑏 = ⟨𝑣, 𝑒⟩) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
4544ex 449 . . . . . . . . . . . . . . 15 (𝑎 = ⟨𝑤, 𝑓⟩ → (𝑏 = ⟨𝑣, 𝑒⟩ → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
4645adantr 480 . . . . . . . . . . . . . 14 ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑏 = ⟨𝑣, 𝑒⟩ → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
4746com12 32 . . . . . . . . . . . . 13 (𝑏 = ⟨𝑣, 𝑒⟩ → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
4847adantr 480 . . . . . . . . . . . 12 ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
4948imp 444 . . . . . . . . . . 11 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
5033, 43, 493imtr4d 283 . . . . . . . . . 10 (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏))
5150ex 449 . . . . . . . . 9 ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏)))
5251exlimivv 1900 . . . . . . . 8 (∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏)))
5352com12 32 . . . . . . 7 ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏)))
5453exlimivv 1900 . . . . . 6 (∃𝑤𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏)))
5554imp 444 . . . . 5 ((∃𝑤𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) ∧ ∃𝑣𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)))) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏))
5621, 24, 55syl2anb 495 . . . 4 ((𝑎𝐺𝑏𝐺) → ((2nd𝑎) = (2nd𝑏) → 𝑎 = 𝑏))
577, 56sylbid 230 . . 3 ((𝑎𝐺𝑏𝐺) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
5857rgen2a 3006 . 2 𝑎𝐺𝑏𝐺 ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)
59 dff13 6552 . 2 (𝐹:𝐺1-1𝑃 ↔ (𝐹:𝐺𝑃 ∧ ∀𝑎𝐺𝑏𝐺 ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
604, 58, 59mpbir2an 975 1 𝐹:𝐺1-1𝑃
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  𝒫 cpw 4191  ⟨cop 4216  {copab 4745   ↦ cmpt 4762  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  2nd c2nd 7209  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-edg 25985  df-upgr 26022  df-uspgr 26090  df-spr 42053 This theorem is referenced by:  uspgrsprf1o  42082
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