Theorem swapf2fn 49755

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 2737	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))
2		ovex 7393	. . . 4 ⊢ (𝑢(Hom ‘𝑆)𝑣) ∈ V
3	2	mptex 7171	. . 3 ⊢ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}) ∈ V
4	1, 3	fnmpoi 8016	. 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 2738	. . . 4 ⊢ (𝜑 → (Hom ‘𝑆) = (Hom ‘𝑆))
10		swapf1val.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
11	5, 6, 7, 8, 9, 10	swapf2fval 49752	. . 3 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})))
12	11	fneq1d 6585	. 2 ⊢ (𝜑 → (𝑃 Fn (𝐵 × 𝐵) ↔ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) Fn (𝐵 × 𝐵)))
13	4, 12	mpbiri 258	1 ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4568  ⟨cop 4574  ∪ cuni 4851   ↦ cmpt 5167   × cxp 5622  ^◡ccnv 5623   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17170  Hom chom 17222   ×_c cxpc 18125   swap_F cswapf 49746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-swapf 49747

This theorem is referenced by:  swapffunc  49769