Theorem swapf2vala 49395

Description: The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapf1a.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
swapf1a.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf1a.b	⊢ 𝐵 = (Base‘𝑆)
swapf1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
swapf2a.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))

Assertion

Ref	Expression
swapf2vala	⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Distinct variable groups:   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑃(𝑓)   𝑆(𝑓)   𝑂(𝑓)

Proof of Theorem swapf2vala

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

Step	Hyp	Ref	Expression
1		swapf1a.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
2		swapf1a.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
3		swapf1a.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4	1, 2, 3	elxpcbasex1 49373	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5	1, 2, 3	elxpcbasex2 49375	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
8	4, 5, 1, 2, 6, 7	swapf2fval 49390	. 2 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
9		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑢 = 𝑋)
10		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑣 = 𝑌)
11	9, 10	oveq12d 7370	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑢𝐻𝑣) = (𝑋𝐻𝑌))
12	11	mpteq1d 5183	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
13		swapf2a.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7385	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
15	14	mptex 7163	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V)
17	8, 12, 3, 13, 16	ovmpod 7504	1 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4575  ⟨cop 4581  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  Hom chom 17174   ×_c cxpc 18076   swap_F cswapf 49384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-1cn 11071  ax-addcl 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-nn 12133  df-slot 17095  df-ndx 17107  df-base 17123  df-xpc 18080  df-swapf 49385

This theorem is referenced by:  swapf2a  49396  swapf2val  49398  swapf2f1oaALT  49403