Theorem swapfval 49013

Description: Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapfval.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapfval.b	⊢ 𝐵 = (Base‘𝑆)
swapfval.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))

Assertion

Ref	Expression
swapfval	⊢ (𝜑 → (𝐶swapF𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)

Distinct variable groups:   𝑢,𝐵,𝑣,𝑥   𝑢,𝐶,𝑣   𝑢,𝐷,𝑣   𝑓,𝐻,𝑢,𝑣   𝑢,𝑆,𝑣   𝜑,𝑢,𝑣   𝑥,𝑓

Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑓)   𝐶(𝑥,𝑓)   𝐷(𝑥,𝑓)   𝑆(𝑥,𝑓)   𝑈(𝑥,𝑣,𝑢,𝑓)   𝐻(𝑥)   𝑉(𝑥,𝑣,𝑢,𝑓)

Proof of Theorem swapfval

Dummy variables 𝑏 𝑐 𝑑 ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-swapf 49011	. . 3 ⊢ 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	1	a1i 11	. 2 ⊢ (𝜑 → 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}))⟩))
3		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) ∈ V)
4		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	oveq12d 7431	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = (𝐶 ×c 𝐷))
7		swapfval.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
8	6, 7	eqtr4di 2787	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 ×c 𝑑) = 𝑆)
9		fvexd 6901	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) ∈ V)
10		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
11	10	fveq2d 6890	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
12		swapfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2787	. . . 4 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → (Base‘𝑠) = 𝐵)
14		fvexd 6901	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) ∈ V)
15		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝑠 = 𝑆)
16	15	fveq2d 6890	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = (Hom ‘𝑆))
17		swapfval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
18	17	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → 𝐻 = (Hom ‘𝑆))
19	16, 18	eqtr4d 2772	. . . . 5 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐻)
20		simplr 768	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵)
21	20	mpteq1d 5217	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
22		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
23	22	oveqd 7430	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑢ℎ𝑣) = (𝑢𝐻𝑣))
24	23	mpteq1d 5217	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
25	20, 20, 24	mpoeq123dv 7490	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
26	21, 25	opeq12d 4861	. . . . 5 ⊢ (((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
27	14, 19, 26	csbied2 3916	. . . 4 ⊢ ((((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) ∧ 𝑏 = 𝐵) → ⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
28	9, 13, 27	csbied2 3916	. . 3 ⊢ (((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
29	3, 8, 28	csbied2 3916	. 2 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩ = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
30		swapfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
31	30	elexd 3487	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
32		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
33	32	elexd 3487	. 2 ⊢ (𝜑 → 𝐷 ∈ V)
34		opex 5449	. . 3 ⊢ ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩ ∈ V
35	34	a1i 11	. 2 ⊢ (𝜑 → ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩ ∈ V)
36	2, 29, 31, 33, 35	ovmpod 7567	1 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⦋csb 3879  {csn 4606  ⟨cop 4612  ∪ cuni 4887   ↦ cmpt 5205  ^◡ccnv 5664  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  Basecbs 17230  Hom chom 17285   ×_c cxpc 18184  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-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-swapf 49011

This theorem is referenced by:  swapfelvv  49014  swapf2fvala  49015  swapf1vala  49017