Theorem dfswapf2 49223

Description: Alternate definition of swap_F (df-swapf 49222). (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 49222	. 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 6853	. . . . . 6 ⊢ (Base‘(𝑐 ×c 𝑑)) ∈ V
3		id 22	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = (Base‘(𝑐 ×c 𝑑)))
4		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑐 ×c 𝑑) = (𝑐 ×c 𝑑)
5		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑐) = (Base‘𝑐)
6		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑑) = (Base‘𝑑)
7	4, 5, 6	xpcbas 18115	. . . . . . . . . 10 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = (Base‘(𝑐 ×c 𝑑))
8	3, 7	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → 𝑏 = ((Base‘𝑐) × (Base‘𝑑)))
9	8	mpteq1d 5192	. . . . . . . 8 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥}))
10		eqidd 2730	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))
11	8, 8, 10	mpoeq123dv 7444	. . . . . . . 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 3866	. . . . . 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 3894	. . . . 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 7402	. . . . . 6 ⊢ (𝑐 ×c 𝑑) ∈ V
16		fveq2 6840	. . . . . . 7 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Base‘𝑠) = (Base‘(𝑐 ×c 𝑑)))
17		fveq2 6840	. . . . . . . 8 ⊢ (𝑠 = (𝑐 ×c 𝑑) → (Hom ‘𝑠) = (Hom ‘(𝑐 ×c 𝑑)))
18	17	csbeq1d 3863	. . . . . . 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 3868	. . . . . 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 3894	. . . . 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 3863	. . . . . . . 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 3868	. . . . . . 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 3894	. . . . . 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 5939	. . . . . . . . 9 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ 𝑏) = (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))))
25		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑏 = (Base‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢ℎ𝑣)))
26	8, 8, 25	mpoeq123dv 7444	. . . . . . . . 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 3866	. . . . . . 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 3894	. . . . . 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 2729	. . . . . . . . 9 ⊢ ((Base‘𝑐) × (Base‘𝑑)) = ((Base‘𝑐) × (Base‘𝑑))
31	30	tposideq2 48850	. . . . . . . 8 ⊢ (tpos I ↾ ((Base‘𝑐) × (Base‘𝑑))) = (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑑)) ↦ ∪ ◡{𝑥})
32		eqid 2729	. . . . . . . . . . 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 48850	. . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑐) = (Hom ‘𝑐)
35		eqid 2729	. . . . . . . . . . . 12 ⊢ (Hom ‘𝑑) = (Hom ‘𝑑)
36		eqid 2729	. . . . . . . . . . . 12 ⊢ (Hom ‘(𝑐 ×c 𝑑)) = (Hom ‘(𝑐 ×c 𝑑))
37		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)) ∧ 𝑣 ∈ ((Base‘𝑐) × (Base‘𝑑))) → 𝑢 ∈ ((Base‘𝑐) × (Base‘𝑑)))
38		simpr 484	. . . . . . . . . . . 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 18117	. . . . . . . . . . 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 5939	. . . . . . . . . 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 5192	. . . . . . . . . 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 2790	. . . . . . . . 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 7447	. . . . . . . 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 6853	. . . . . . . 8 ⊢ (Hom ‘(𝑐 ×c 𝑑)) ∈ V
46		oveq 7375	. . . . . . . . . . 11 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑢ℎ𝑣) = (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣))
47	46	reseq2d 5939	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (tpos I ↾ (𝑢ℎ𝑣)) = (tpos I ↾ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣)))
48	47	mpoeq3dv 7448	. . . . . . . . 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 3894	. . . . . . 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 5192	. . . . . . . . . 10 ⊢ (ℎ = (Hom ‘(𝑐 ×c 𝑑)) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢(Hom ‘(𝑐 ×c 𝑑))𝑣) ↦ ∪ ◡{𝑓}))
52	51	mpoeq3dv 7448	. . . . . . . . 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 3894	. . . . . . 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 2762	. . . . . 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 2756	. . . . 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 2763	. . . 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 7447	. 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 2755	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)))⟩)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⦋csb 3859  {csn 4585  ⟨cop 4591  ∪ cuni 4867   ↦ cmpt 5183   I cid 5525   × cxp 5629  ^◡ccnv 5630   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  tpos ctpos 8181  Basecbs 17155  Hom chom 17207   ×_c cxpc 18105   swap_F cswapf 49221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-xpc 18109  df-swapf 49222

This theorem is referenced by: (None)