Theorem swapf2fn 50122

Description: The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapf2fvala.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf2fvala.b	⊢ 𝐵 = (Base‘𝑆)
swapf1val.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)

Assertion

Ref	Expression
swapf2fn	⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))

Proof of Theorem swapf2fn

Dummy variables 𝑢 𝑣 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))
2		ovex 7452	. . . 4 ⊢ (𝑢(Hom ‘𝑆)𝑣) ∈ V
3	2	mptex 7228	. . 3 ⊢ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}) ∈ V
4	1, 3	fnmpoi 8073	. 2 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) Fn (𝐵 × 𝐵)
5		swapfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6		swapfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
7		swapf2fvala.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
8		swapf2fvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
9		eqidd 2766	. . . 4 ⊢ (𝜑 → (Hom ‘𝑆) = (Hom ‘𝑆))
10		swapf1val.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
11	5, 6, 7, 8, 9, 10	swapf2fval 50119	. . 3 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})))
12	11	fneq1d 6632	. 2 ⊢ (𝜑 → (𝑃 Fn (𝐵 × 𝐵) ↔ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) Fn (𝐵 × 𝐵)))
13	4, 12	mpbiri 261	1 ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {csn 4591  ⟨cop 4597  ∪ cuni 4874   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Basecbs 17295  Hom chom 17347   ×_c cxpc 18250   swap_F cswapf 50113

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-swapf 50114

This theorem is used by:  swapffunc  50136