Theorem tposcurf1 49044

Description: Value of the object part of the transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
tposcurf1.g	⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))
tposcurf1.a	⊢ 𝐴 = (Base‘𝐶)
tposcurf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
tposcurf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
tposcurf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
tposcurf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposcurf1.k	⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
tposcurf1.b	⊢ 𝐵 = (Base‘𝐷)
tposcurf1.j	⊢ 𝐽 = (Hom ‘𝐷)
tposcurf1.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
tposcurf1	⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

Distinct variable groups:   1 ,𝑔,𝑦,𝑧   𝑦,𝐴   𝐵,𝑔,𝑦,𝑧   𝐶,𝑔,𝑦,𝑧   𝐷,𝑔,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐹,𝑦,𝑧   𝑔,𝐽   𝑔,𝑋,𝑦,𝑧   𝜑,𝑔,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)   𝐾(𝑦,𝑧,𝑔)

Proof of Theorem tposcurf1

Step	Hyp	Ref	Expression
1		tposcurf1.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
2		tposcurf1.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))
3	2	fveq2d 6890	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷)))))
4	3	fveq1d 6888	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))‘𝑋))
5		eqid 2734	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))) = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷)))
6		tposcurf1.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
7		tposcurf1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		tposcurf1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9		tposcurf1.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
10		eqidd 2735	. . . . 5 ⊢ (𝜑 → (𝐹 ∘func (𝐶swapF𝐷)) = (𝐹 ∘func (𝐶swapF𝐷)))
11	7, 8, 9, 10	cofuswapfcl 49038	. . . 4 ⊢ (𝜑 → (𝐹 ∘func (𝐶swapF𝐷)) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
12		tposcurf1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
13		tposcurf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14		eqid 2734	. . . 4 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))‘𝑋)
15		tposcurf1.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
16		tposcurf1.1	. . . 4 ⊢ 1 = (Id‘𝐶)
17	5, 6, 7, 8, 11, 12, 13, 14, 15, 16	curf1 18241	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))‘𝑋) = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
18	1, 4, 17	3eqtrd 2773	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
19	12	fvexi 6900	. . . . . . . . 9 ⊢ 𝐵 ∈ V
20	19	mptex 7225	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)) ∈ V
21	19, 19	mpoex 8086	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔))) ∈ V
22	20, 21	op1std 8006	. . . . . . 7 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)))
23	18, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)))
24		ovexd 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦) ∈ V)
25	23, 24	fvmpt2d 7009	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦))
26	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))
27	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
28	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
29	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
30	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
31	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐾 = ((1st ‘𝐺)‘𝑋))
32		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
33	26, 6, 27, 28, 29, 30, 31, 12, 32	tposcurf11 49042	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑦(1st ‘𝐹)𝑋))
34	25, 33	eqtr3d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦) = (𝑦(1st ‘𝐹)𝑋))
35	34	mpteq2dva 5222	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)))
36	20, 21	op2ndd 8007	. . . . . . . . . 10 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
37	18, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
38		ovex 7446	. . . . . . . . . . 11 ⊢ (𝑦𝐽𝑧) ∈ V
39	38	mptex 7225	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V
40	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
41	37, 40	ovmpt4d 48749	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))
42		ovexd 7448	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔) ∈ V)
43	41, 42	fvmpt2d 7009	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔))
44	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶swapF𝐷))))
45	7	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐶 ∈ Cat)
46	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐷 ∈ Cat)
47	9	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
48	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑋 ∈ 𝐴)
49	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐾 = ((1st ‘𝐺)‘𝑋))
50		simplrl 776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑦 ∈ 𝐵)
51		simplrr 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑧 ∈ 𝐵)
52		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑔 ∈ (𝑦𝐽𝑧))
53	44, 6, 45, 46, 47, 48, 49, 12, 50, 15, 16, 51, 52	tposcurf12 49043	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))
54	43, 53	eqtr3d 2771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔) = (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))
55	54	mpteq2dva 5222	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
56	55	3impb 1114	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
57	56	mpoeq3dva 7492	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))))
58	35, 57	opeq12d 4861	. 2 ⊢ (𝜑 → ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶swapF𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)
59	18, 58	eqtrd 2769	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612   ↦ cmpt 5205  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994  2^nd c2nd 7995  Basecbs 17230  Hom chom 17285  Catccat 17679  Idccid 17680   Func cfunc 17871   ∘_func ccofu 17873   ×_c cxpc 18184   curry_F ccurf 18226  swap_Fcswapf 49010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-xpc 18188  df-curf 18230  df-swapf 49011

This theorem is referenced by:  precofval  49112  precofvalALT  49113