Theorem dfswapf2 50338

Description: Alternate definition of swap_F (df-swapf 50337). (Contributed by Zhi Wang, 9-Oct-2025.)

Assertion

Ref	Expression
dfswapf2	⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)

Distinct variable group:   𝑏,𝑐,𝑑,ℎ,𝑠,𝑢,𝑣

Proof of Theorem dfswapf2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-swapf 50337	. 2 ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
2		fvex 6896	. . . . . 6 ⊢ (Base‘(𝑐 ×c 𝑑)) ∈ V
3		id 23	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = (Base‘(𝑐 ×c 𝑑)))
4		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑐 ×c 𝑑) = (𝑐 ×c 𝑑)
5		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑐) = (Base‘𝑐)
6		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑑) = (Base‘𝑑)
7	4, 5, 6	xpcbas 18345	. . . . . . . . . 10 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = (Base‘(𝑐 ×c 𝑑))
8	3, 7	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = ((Base‘𝑐) × (Base‘𝑑)))
9	8	mpteq1d 5195	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}))
10		eqidd 2762	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))
11	8, 8, 10	mpoeq123dv 7493	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})))
12	9, 11	opeq12d 4841	. . . . . . 7 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
13	12	csbeq2dv 3854	. . . . . 6 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
14	2, 13	csbie 3882	. . . . 5 ⊢ ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
15		ovex 7451	. . . . . 6 ⊢ (𝑐 ×c 𝑑) ∈ V
16		fveq2 6883	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Base‘𝑠) = (Base‘(𝑐 ×c 𝑑)))
17		fveq2 6883	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Hom ‘𝑠) = (Hom ‘(𝑐 ×c 𝑑)))
18	17	csbeq1d 3851	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
19	16, 18	csbeq12dv 3856	. . . . . 6 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
20	15, 19	csbie 3882	. . . . 5 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
21	17	csbeq1d 3851	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
22	16, 21	csbeq12dv 3856	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
23	15, 22	csbie 3882	. . . . . 6 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩
24	8	reseq2d 5970	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ 𝑏) = (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))))
25		eqidd 2762	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢ℎ𝑣)))
26	8, 8, 25	mpoeq123dv 7493	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣))))
27	24, 26	opeq12d 4841	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
28	27	csbeq2dv 3854	. . . . . . 7 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
29	2, 28	csbie 3882	. . . . . 6 ⊢ ⦋(Base‘(𝑐 ×c 𝑑)) / 𝑏⦌⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩
30		eqid 2761	. . . . . . . . 9 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝑐) × (Base‘𝑑))
31	30	tposideq2 49966	. . . . . . . 8 ⊢ (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥})
32		eqid 2761	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) = (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))
33	32	tposideq2 49966	. . . . . . . . . 10 ⊢ (tpos I ↾ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))) = (𝑓 ∈ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) ↦ ∪ ◡{𝑓})
34		eqid 2761	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑐) = (Hom ‘𝑐)
35		eqid 2761	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑑) = (Hom ‘𝑑)
36		eqid 2761	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘(𝑐 ×c 𝑑))
37		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)))
38		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)))
39	4, 7, 34, 35, 36, 37, 38	xpchom 18347	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) = (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))))
40	39	reseq2d 5970	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)) = (tpos I ↾ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣)))))
41	39	mpteq1d 5195	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑢)(Hom ‘𝑐)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑑)(2nd ‘𝑣))) ↦ ∪ ◡{𝑓}))
42	33, 40, 41	3eqtr4a 2822	. . . . . . . . 9 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
43	42	mpoeq3ia 7496	. . . . . . . 8 ⊢ (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
44	31, 43	opeq12i 4838	. . . . . . 7 ⊢ ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩
45		fvex 6896	. . . . . . . 8 ⊢ (Hom ‘(𝑐 ×c 𝑑)) ∈ V
46		oveq 7424	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢ℎ𝑣) = (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))
47	46	reseq2d 5970	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))
48	47	mpoeq3dv 7497	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣))) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))))
49	48	opeq2d 4840	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩)
50	45, 49	csbie 3882	. . . . . . 7 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))⟩
51	46	mpteq1d 5195	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
52	51	mpoeq3dv 7497	. . . . . . . . 9 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓})))
53	52	opeq2d 4840	. . . . . . . 8 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩)
54	45, 53	csbie 3882	. . . . . . 7 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))⟩
55	44, 50, 54	3eqtr4i 2794	. . . . . 6 ⊢ ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
56	23, 29, 55	3eqtri 2788	. . . . 5 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(Hom ‘(𝑐 ×c 𝑑)) / ℎ⦌⟨(𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)), 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
57	14, 20, 56	3eqtr4ri 2795	. . . 4 ⊢ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩
58	57	a1i 11	. . 3 ⊢ ((𝑐 ∈ V ∧ 𝑑 ∈ V) → ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩ = ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
59	58	mpoeq3ia 7496	. 2 ⊢ (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩) = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
60	1, 59	eqtr4i 2787	1 ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⦋csb 3847  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186   I cid 5545   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  2^nd c2nd 7998  tpos ctpos 8235  Basecbs 17380  Hom chom 17432   ×_c cxpc 18335   swap_F cswapf 50336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-xpc 18339  df-swapf 50337

This theorem is used by: (None)