Theorem swapf2a 50348

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))
swapf2a.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
swapf2a	⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

Proof of Theorem swapf2a

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapf1a.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
3		swapf1a.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
4		swapf1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		swapf2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7	1, 2, 3, 4, 5, 6	swapf2vala 50347	. 2 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
8		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9	8	sneqd 4596	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → {𝑓} = {𝐹})
10	9	cnveqd 5853	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡{𝑓} = ◡{𝐹})
11	10	unieqd 4880	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ∪ ◡{𝐹})
12		swapf2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
13	6	oveqd 7435	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝑆)𝑌))
14		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
17	2, 3, 14, 15, 16, 4, 5	xpchom 18347	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝑆)𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
18	13, 17	eqtrd 2796	. . . . . 6 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
19	12, 18	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
20		2nd1st 8047	. . . . 5 ⊢ (𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
21	19, 20	syl 18	. . . 4 ⊢ (𝜑 → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
23	11, 22	eqtrd 2796	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
24		opex 5432	. . 3 ⊢ ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V)
26	7, 23, 12, 25	fvmptd 6999	1 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590  ∪ cuni 4867   × cxp 5649  ^◡ccnv 5650  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380  Hom chom 17432   ×_c cxpc 18335   swap_F cswapf 50336

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-xpc 18339  df-swapf 50337

This theorem is used by:  swapfcoa  50358