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Theorem uspgrsprfo 43519
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 43517 . . 3 𝐹:𝐺𝑃
54a1i 11 . 2 (𝑉𝑊𝐹:𝐺𝑃)
61eleq2i 2873 . . . . . . 7 (𝑎𝑃𝑎 ∈ 𝒫 (Pairs‘𝑉))
7 selpw 4462 . . . . . . 7 (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
86, 7bitri 276 . . . . . 6 (𝑎𝑃𝑎 ⊆ (Pairs‘𝑉))
9 eqidd 2795 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑉 = 𝑉)
10 vex 3439 . . . . . . . . . . . . . . 15 𝑎 ∈ V
1110a1i 11 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ∈ V)
12 f1oi 6523 . . . . . . . . . . . . . . . . 17 ( I ↾ 𝑎):𝑎1-1-onto𝑎
1312a1i 11 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):𝑎1-1-onto𝑎)
14 dmresi 5802 . . . . . . . . . . . . . . . . 17 dom ( I ↾ 𝑎) = 𝑎
15 f1oeq2 6476 . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎)
18 sprvalpwle2 43147 . . . . . . . . . . . . . . . . 17 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
1918sseq2d 3922 . . . . . . . . . . . . . . . 16 (𝑉𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2019biimpac 479 . . . . . . . . . . . . . . 15 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
2117, 20jca 512 . . . . . . . . . . . . . 14 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22 f1oeq3 6477 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑎))
23 sseq1 3915 . . . . . . . . . . . . . . 15 (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2422, 23anbi12d 630 . . . . . . . . . . . . . 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 3605 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto𝑓𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
26 resiexg 7478 . . . . . . . . . . . . . . 15 (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
2710, 26ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ 𝑎) ∈ V
2827f11o 7507 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 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 6848 . . . . . . . . . . . . . . 15 (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
3231anim2i 616 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
3332ancoms 459 . . . . . . . . . . . . 13 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V))
34 isuspgrop 26629 . . . . . . . . . . . . 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 258 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
37 fveqeq2 6550 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
38 fveqeq2 6550 . . . . . . . . . . . . 13 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
3937, 38anbi12d 630 . . . . . . . . . . . 12 (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
4039adantl 482 . . . . . . . . . . 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 26472 . . . . . . . . . . . . . 14 ((𝑉𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
4232, 41syl 17 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
43 edgopval 26519 . . . . . . . . . . . . . . 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 5822 . . . . . . . . . . . . . 14 ran ( I ↾ 𝑎) = 𝑎
4644, 45syl6eq 2846 . . . . . . . . . . . . 13 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
4742, 46jca 512 . . . . . . . . . . . 12 ((𝑉𝑊𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4847ancoms 459 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
4936, 40, 48rspcedvd 3564 . . . . . . . . . 10 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
509, 49jca 512 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
512eleq2i 2873 . . . . . . . . . 10 (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
5230anim1i 614 . . . . . . . . . . . 12 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑎 ∈ V ∧ 𝑉𝑊))
5352ancomd 462 . . . . . . . . . . 11 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (𝑉𝑊𝑎 ∈ V))
54 eqeq1 2798 . . . . . . . . . . . . . 14 (𝑣 = 𝑉 → (𝑣 = 𝑉𝑉 = 𝑉))
5554adantr 481 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (𝑣 = 𝑉𝑉 = 𝑉))
56 eqeq2 2805 . . . . . . . . . . . . . . 15 (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
57 eqeq2 2805 . . . . . . . . . . . . . . 15 (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
5856, 57bi2anan9 635 . . . . . . . . . . . . . 14 ((𝑣 = 𝑉𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
5958rexbidv 3259 . . . . . . . . . . . . 13 ((𝑣 = 𝑉𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
6055, 59anbi12d 630 . . . . . . . . . . . 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 5313 . . . . . . . . . . 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 284 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
6450, 63mpbird 258 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
65 fveq2 6541 . . . . . . . . . 10 (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
6665eqeq2d 2804 . . . . . . . . 9 (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
6766adantl 482 . . . . . . . 8 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
68 op2ndg 7561 . . . . . . . . . . 11 ((𝑉𝑊𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
6968elvd 3442 . . . . . . . . . 10 (𝑉𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7069adantl 482 . . . . . . . . 9 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
7170eqcomd 2800 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
7264, 67, 71rspcedvd 3564 . . . . . . 7 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉𝑊) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
7372ex 413 . . . . . 6 (𝑎 ⊆ (Pairs‘𝑉) → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
748, 73sylbi 218 . . . . 5 (𝑎𝑃 → (𝑉𝑊 → ∃𝑏𝐺 𝑎 = (2nd𝑏)))
7574impcom 408 . . . 4 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (2nd𝑏))
761, 2, 3uspgrsprfv 43516 . . . . . . 7 (𝑏𝐺 → (𝐹𝑏) = (2nd𝑏))
7776adantl 482 . . . . . 6 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝐹𝑏) = (2nd𝑏))
7877eqeq2d 2804 . . . . 5 (((𝑉𝑊𝑎𝑃) ∧ 𝑏𝐺) → (𝑎 = (𝐹𝑏) ↔ 𝑎 = (2nd𝑏)))
7978rexbidva 3258 . . . 4 ((𝑉𝑊𝑎𝑃) → (∃𝑏𝐺 𝑎 = (𝐹𝑏) ↔ ∃𝑏𝐺 𝑎 = (2nd𝑏)))
8075, 79mpbird 258 . . 3 ((𝑉𝑊𝑎𝑃) → ∃𝑏𝐺 𝑎 = (𝐹𝑏))
8180ralrimiva 3148 . 2 (𝑉𝑊 → ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏))
82 dffo3 6734 . 2 (𝐹:𝐺onto𝑃 ↔ (𝐹:𝐺𝑃 ∧ ∀𝑎𝑃𝑏𝐺 𝑎 = (𝐹𝑏)))
835, 81, 82sylanbrc 583 1 (𝑉𝑊𝐹:𝐺onto𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1762  wcel 2080  wral 3104  wrex 3105  {crab 3108  Vcvv 3436  cdif 3858  wss 3861  c0 4213  𝒫 cpw 4455  {csn 4474  cop 4480   class class class wbr 4964  {copab 5026  cmpt 5043   I cid 5350  dom cdm 5446  ran crn 5447  cres 5448  wf 6224  1-1wf1 6225  ontowfo 6226  1-1-ontowf1o 6227  cfv 6228  2nd c2nd 7547  cle 10525  2c2 11542  chash 13540  Vtxcvtx 26464  Edgcedg 26515  USPGraphcuspgr 26616  Pairscspr 43135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-2o 7957  df-oadd 7960  df-er 8142  df-en 8361  df-dom 8362  df-sdom 8363  df-fin 8364  df-dju 9179  df-card 9217  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-nn 11489  df-2 11550  df-n0 11748  df-xnn0 11818  df-z 11832  df-uz 12094  df-fz 12743  df-hash 13541  df-vtx 26466  df-iedg 26467  df-edg 26516  df-upgr 26550  df-uspgr 26618  df-spr 43136
This theorem is referenced by:  uspgrsprf1o  43520
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