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Theorem fucval 17910
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucval.q 𝑄 = (𝐢 FuncCat 𝐷)
fucval.b 𝐡 = (𝐢 Func 𝐷)
fucval.n 𝑁 = (𝐢 Nat 𝐷)
fucval.a 𝐴 = (Baseβ€˜πΆ)
fucval.o Β· = (compβ€˜π·)
fucval.c (πœ‘ β†’ 𝐢 ∈ Cat)
fucval.d (πœ‘ β†’ 𝐷 ∈ Cat)
fucval.x (πœ‘ β†’ βˆ™ = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
Assertion
Ref Expression
fucval (πœ‘ β†’ 𝑄 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩})
Distinct variable groups:   𝑣,β„Ž,𝐡   π‘Ž,𝑏,𝑓,𝑔,β„Ž,𝑣,π‘₯,πœ‘   𝐢,π‘Ž,𝑏,𝑓,𝑔,β„Ž,𝑣,π‘₯   𝐷,π‘Ž,𝑏,𝑓,𝑔,β„Ž,𝑣,π‘₯
Allowed substitution hints:   𝐴(π‘₯,𝑣,𝑓,𝑔,β„Ž,π‘Ž,𝑏)   𝐡(π‘₯,𝑓,𝑔,π‘Ž,𝑏)   𝑄(π‘₯,𝑣,𝑓,𝑔,β„Ž,π‘Ž,𝑏)   βˆ™ (π‘₯,𝑣,𝑓,𝑔,β„Ž,π‘Ž,𝑏)   Β· (π‘₯,𝑣,𝑓,𝑔,β„Ž,π‘Ž,𝑏)   𝑁(π‘₯,𝑣,𝑓,𝑔,β„Ž,π‘Ž,𝑏)

Proof of Theorem fucval
Dummy variables 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucval.q . 2 𝑄 = (𝐢 FuncCat 𝐷)
2 df-fuc 17895 . . . 4 FuncCat = (𝑑 ∈ Cat, 𝑒 ∈ Cat ↦ {⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩, ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩})
32a1i 11 . . 3 (πœ‘ β†’ FuncCat = (𝑑 ∈ Cat, 𝑒 ∈ Cat ↦ {⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩, ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩}))
4 simprl 770 . . . . . . 7 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ 𝑑 = 𝐢)
5 simprr 772 . . . . . . 7 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ 𝑒 = 𝐷)
64, 5oveq12d 7427 . . . . . 6 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑑 Func 𝑒) = (𝐢 Func 𝐷))
7 fucval.b . . . . . 6 𝐡 = (𝐢 Func 𝐷)
86, 7eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑑 Func 𝑒) = 𝐡)
98opeq2d 4881 . . . 4 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩ = ⟨(Baseβ€˜ndx), 𝐡⟩)
104, 5oveq12d 7427 . . . . . 6 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑑 Nat 𝑒) = (𝐢 Nat 𝐷))
11 fucval.n . . . . . 6 𝑁 = (𝐢 Nat 𝐷)
1210, 11eqtr4di 2791 . . . . 5 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑑 Nat 𝑒) = 𝑁)
1312opeq2d 4881 . . . 4 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩ = ⟨(Hom β€˜ndx), π‘βŸ©)
148sqxpeqd 5709 . . . . . . 7 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)) = (𝐡 Γ— 𝐡))
1512oveqd 7426 . . . . . . . . . 10 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑔(𝑑 Nat 𝑒)β„Ž) = (π‘”π‘β„Ž))
1612oveqd 7426 . . . . . . . . . 10 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑓(𝑑 Nat 𝑒)𝑔) = (𝑓𝑁𝑔))
174fveq2d 6896 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (Baseβ€˜π‘‘) = (Baseβ€˜πΆ))
18 fucval.a . . . . . . . . . . . 12 𝐴 = (Baseβ€˜πΆ)
1917, 18eqtr4di 2791 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (Baseβ€˜π‘‘) = 𝐴)
205fveq2d 6896 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (compβ€˜π‘’) = (compβ€˜π·))
21 fucval.o . . . . . . . . . . . . . 14 Β· = (compβ€˜π·)
2220, 21eqtr4di 2791 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (compβ€˜π‘’) = Β· )
2322oveqd 7426 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯)) = (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯)))
2423oveqd 7426 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) = ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))
2519, 24mpteq12dv 5240 . . . . . . . . . 10 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) = (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))
2615, 16, 25mpoeq123dv 7484 . . . . . . . . 9 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) = (𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))
2726csbeq2dv 3901 . . . . . . . 8 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) = ⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))
2827csbeq2dv 3901 . . . . . . 7 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) = ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))
2914, 8, 28mpoeq123dv 7484 . . . . . 6 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))) = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
30 fucval.x . . . . . . 7 (πœ‘ β†’ βˆ™ = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
3130adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ βˆ™ = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
3229, 31eqtr4d 2776 . . . . 5 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))) = βˆ™ )
3332opeq2d 4881 . . . 4 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩ = ⟨(compβ€˜ndx), βˆ™ ⟩)
349, 13, 33tpeq123d 4753 . . 3 ((πœ‘ ∧ (𝑑 = 𝐢 ∧ 𝑒 = 𝐷)) β†’ {⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩, ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩})
35 fucval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
36 fucval.d . . 3 (πœ‘ β†’ 𝐷 ∈ Cat)
37 tpex 7734 . . . 4 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩} ∈ V
3837a1i 11 . . 3 (πœ‘ β†’ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩} ∈ V)
393, 34, 35, 36, 38ovmpod 7560 . 2 (πœ‘ β†’ (𝐢 FuncCat 𝐷) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩})
401, 39eqtrid 2785 1 (πœ‘ β†’ 𝑄 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  β¦‹csb 3894  {ctp 4633  βŸ¨cop 4635   ↦ cmpt 5232   Γ— cxp 5675  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  ndxcnx 17126  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608   Func cfunc 17804   Nat cnat 17892   FuncCat cfuc 17893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fuc 17895
This theorem is referenced by:  fuccofval  17911  fucbas  17912  fuchom  17913  fuchomOLD  17914  fucpropd  17930  catcfuccl  18069  catcfucclOLD  18070
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