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Theorem uspgrsprfo 42428
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 42426 . . 3 𝐹:𝐺𝑃
54a1i 11 . 2 (𝑉𝑊𝐹:𝐺𝑃)
61eleq2i 2836 . . . . . . 7 (𝑎𝑃𝑎 ∈ 𝒫 (Pairs‘𝑉))
7 selpw 4324 . . . . . . 7 (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
86, 7bitri 266 . . . . . 6 (𝑎𝑃𝑎 ⊆ (Pairs‘𝑉))
9 eqidd 2766 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑉 = 𝑉)
10 vex 3353 . . . . . . . . . . . . . . 15 𝑎 ∈ V
1110a1i 11 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ∈ V)
12 f1oi 6359 . . . . . . . . . . . . . . . . 17 ( I ↾ 𝑎):𝑎1-1-onto𝑎
1312a1i 11 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):𝑎1-1-onto𝑎)
14 dmresi 5643 . . . . . . . . . . . . . . . . 17 dom ( I ↾ 𝑎) = 𝑎
15 f1oeq2 6313 . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎)
18 sprvalpwle2 42411 . . . . . . . . . . . . . . . . 17 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
1918sseq2d 3795 . . . . . . . . . . . . . . . 16 (𝑉𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2019biimpac 470 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
2117, 20jca 507 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22 f1oeq3 6314 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎))
23 sseq1 3788 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2422, 23anbi12d 624 . . . . . . . . . . . . . 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 3509 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
26 resiexg 7302 . . . . . . . . . . . . . . 15 (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
2710, 26ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ 𝑎) ∈ V
2827f11o 7328 . . . . . . . . . . . . 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 225 . . . . . . . . . . . 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 6675 . . . . . . . . . . . . . . 15 (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
3231anim2i 610 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
3332ancoms 450 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
34 isuspgrop 26337 . . . . . . . . . . . . 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 248 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
37 fveqeq2 6386 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
38 fveqeq2 6386 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
3937, 38anbi12d 624 . . . . . . . . . . . 12 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
4039adantl 473 . . . . . . . . . . 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)))
41 opvtxfv 26176 . . . . . . . . . . . . . 14 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
4232, 41syl 17 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
43 edgopval 26224 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
4432, 43syl 17 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
45 rnresi 5663 . . . . . . . . . . . . . 14 ran ( I ↾ 𝑎) = 𝑎
4644, 45syl6eq 2815 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
4742, 46jca 507 . . . . . . . . . . . 12 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4847ancoms 450 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4936, 40, 48rspcedvd 3469 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
509, 49jca 507 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
512eleq2i 2836 . . . . . . . . . 10 (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
5230anim1i 608 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑎 ∈ V ∧ 𝑉𝑊))
5352ancomd 453 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊𝑎 ∈ V))
54 eqeq1 2769 . . . . . . . . . . . . . 14 (𝑣 = 𝑉 → (𝑣 = 𝑉𝑉 = 𝑉))
5554adantr 472 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (𝑣 = 𝑉𝑉 = 𝑉))
56 eqeq2 2776 . . . . . . . . . . . . . . 15 (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
57 eqeq2 2776 . . . . . . . . . . . . . . 15 (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
5856, 57bi2anan9 629 . . . . . . . . . . . . . 14 ((𝑣 = 𝑉𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
5958rexbidv 3199 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
6055, 59anbi12d 624 . . . . . . . . . . . 12 ((𝑣 = 𝑉𝑒 = 𝑎) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6160opelopabga 5151 . . . . . . . . . . 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))))
6253, 61syl 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))))
6351, 62syl5bb 274 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6450, 63mpbird 248 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
65 fveq2 6377 . . . . . . . . . 10 (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
6665eqeq2d 2775 . . . . . . . . 9 (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
6766adantl 473 . . . . . . . 8 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
68 op2ndg 7381 . . . . . . . . . . 11 ((𝑉𝑊𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
6968elvd 3355 . . . . . . . . . 10 (𝑉𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7069adantl 473 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7170eqcomd 2771 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
7264, 67, 71rspcedvd 3469 . . . . . . 7 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
7372ex 401 . . . . . 6 (𝑎 ⊆ (Pairs‘𝑉) → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
748, 73sylbi 208 . . . . 5 (𝑎𝑃 → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
7574impcom 396 . . . 4 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
761, 2, 3uspgrsprfv 42425 . . . . . . 7 (𝑏𝐺 → (𝐹𝑏) = (2nd𝑏))
7776adantl 473 . . . . . 6 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝐹𝑏) = (2nd𝑏))
7877eqeq2d 2775 . . . . 5 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (2nd𝑏)))
7978rexbidva 3196 . . . 4 ((𝑉𝑊𝑎𝑃) → (∃𝑏𝐺 𝑎 = (𝐹𝑏) ↔ ∃𝑏𝐺 𝑎 = (2nd𝑏)))
8075, 79mpbird 248 . . 3 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (𝐹𝑏))
8180ralrimiva 3113 . 2 (𝑉𝑊 → ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏))
82 dffo3 6566 . 2 (𝐹:𝐺onto𝑃 ↔ (𝐹:𝐺𝑃 ∧ ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏)))
835, 81, 82sylanbrc 578 1 (𝑉𝑊𝐹:𝐺onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336  cop 4342   class class class wbr 4811  {copab 4873  cmpt 4890   I cid 5186  dom cdm 5279  ran crn 5280  cres 5281  wf 6066  1-1wf1 6067  ontowfo 6068  1-1-ontowf1o 6069  cfv 6070  2nd c2nd 7367  cle 10331  2c2 11329  chash 13324  Vtxcvtx 26168  Edgcedg 26219  USPGraphcuspgr 26324  Pairscspr 42399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-n0 11541  df-xnn0 11613  df-z 11627  df-uz 11890  df-fz 12537  df-hash 13325  df-vtx 26170  df-iedg 26171  df-edg 26220  df-upgr 26257  df-uspgr 26326  df-spr 42400
This theorem is referenced by:  uspgrsprf1o  42429
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