Theorem tposcurf1 50105

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 (𝐶 swapF 𝐷))))
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 (𝐶 swapF 𝐷))))
3	2	fveq2d 6885	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))))
4	3	fveq1d 6883	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))
5		eqid 2762	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))) = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))
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 2763	. . . . 5 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) = (𝐹 ∘func (𝐶 swapF 𝐷)))
11	7, 8, 9, 10	cofuswapfcl 50099	. . . 4 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
12		tposcurf1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
13		tposcurf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14		eqid 2762	. . . 4 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘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 18287	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋) = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
18	1, 4, 17	3eqtrd 2801	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩)
19	12	fvexi 6895	. . . . . . . . 9 ⊢ 𝐵 ∈ V
20	19	mptex 7221	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) ∈ V
21	19, 19	mpoex 8074	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))) ∈ V
22	20, 21	op1std 7994	. . . . . . 7 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)))
23	18, 22	syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)))
24		ovexd 7447	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) ∈ V)
25	23, 24	fvmpt2d 7003	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦))
26	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
27	7	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
28	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
29	9	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
30	13	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
31	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐾 = ((1st ‘𝐺)‘𝑋))
32		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
33	26, 6, 27, 28, 29, 30, 31, 12, 32	tposcurf11 50103	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑦(1st ‘𝐹)𝑋))
34	25, 33	eqtr3d 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦) = (𝑦(1st ‘𝐹)𝑋))
35	34	mpteq2dva 5203	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)))
36	20, 21	op2ndd 7995	. . . . . . . . . 10 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
37	18, 36	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))))
38		ovex 7445	. . . . . . . . . . 11 ⊢ (𝑦𝐽𝑧) ∈ V
39	38	mptex 7221	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V
40	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
41	37, 40	ovmpt4d 49671	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)))
42		ovexd 7447	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔) ∈ V)
43	41, 42	fvmpt2d 7003	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))
44	2	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
45	7	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐶 ∈ Cat)
46	8	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐷 ∈ Cat)
47	9	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
48	13	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑋 ∈ 𝐴)
49	1	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝐾 = ((1st ‘𝐺)‘𝑋))
50		simplrl 788	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑦 ∈ 𝐵)
51		simplrr 789	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → 𝑧 ∈ 𝐵)
52		simpr 489	. . . . . . . 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 50104	. . . . . . 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 2799	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦𝐽𝑧)) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔) = (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))
55	54	mpteq2dva 5203	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
56	55	3impb 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))
57	56	mpoeq3dva 7489	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋)))))
58	35, 57	opeq12d 4845	. 2 ⊢ (𝜑 → ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨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 2797	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ⟨cop 4594   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327  Catccat 17726  Idccid 17727   Func cfunc 17917   ∘_func ccofu 17919   ×_c cxpc 18230   curry_F ccurf 18272   swap_F cswapf 50065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-slot 17248  df-ndx 17260  df-base 17276  df-hom 17340  df-cco 17341  df-cat 17730  df-cid 17731  df-func 17921  df-cofu 17923  df-xpc 18234  df-curf 18276  df-swapf 50066

This theorem is used by:  precofval  50173  precofvalALT  50174