Theorem cofuswapf2 49456

Description: The morphism part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
cofuswapf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
cofuswapf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
cofuswapf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
cofuswapf1.g	⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
cofuswapf1.a	⊢ 𝐴 = (Base‘𝐶)
cofuswapf1.b	⊢ 𝐵 = (Base‘𝐷)
cofuswapf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
cofuswapf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofuswapf2.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)
cofuswapf2.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
cofuswapf2.h	⊢ 𝐻 = (Hom ‘𝐶)
cofuswapf2.j	⊢ 𝐽 = (Hom ‘𝐷)
cofuswapf2.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑍))
cofuswapf2.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐽𝑊))

Assertion

Ref	Expression
cofuswapf2	⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))

Proof of Theorem cofuswapf2

Step	Hyp	Ref	Expression
1		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
2	1	fveq2d 6835	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
3	2	oveqd 7372	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩))
4	3	oveqd 7372	. . 3 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩)𝑁))
5		df-ov 7358	. . . 4 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
6		eqid 2733	. . . . . 6 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
7		cofuswapf1.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
8		cofuswapf1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
9	6, 7, 8	xpcbas 18092	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2733	. . . . . 6 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 49445	. . . . 5 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ ((𝐶 ×c 𝐷) Func (𝐷 ×c 𝐶)))
14		cofuswapf1.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
15		cofuswapf1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		cofuswapf1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	15, 16	opelxpd 5660	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18		cofuswapf2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
19		cofuswapf2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
20	18, 19	opelxpd 5660	. . . . 5 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
21		eqid 2733	. . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
22		cofuswapf2.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑍))
23		cofuswapf2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐽𝑊))
24	22, 23	opelxpd 5660	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
25		cofuswapf2.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
26		cofuswapf2.j	. . . . . . 7 ⊢ 𝐽 = (Hom ‘𝐷)
27	6, 7, 8, 25, 26, 15, 16, 18, 19, 21	xpchom2 18100	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
28	24, 27	eleqtrrd 2836	. . . . 5 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
29	9, 13, 14, 17, 20, 21, 28	cofu2 17801	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ((((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)))
30	5, 29	eqtrid 2780	. . 3 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩)𝑁) = ((((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)))
31		df-ov 7358	. . . . . 6 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
32	10, 11	swapfelvv 49424	. . . . . . . 8 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
33		1st2nd2 7969	. . . . . . . 8 ⊢ ((𝐶 swapF 𝐷) ∈ (V × V) → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
35	15, 7	eleqtrdi 2843	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
36	16, 8	eleqtrdi 2843	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
37	34, 35, 36	swapf1 49433	. . . . . 6 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
38	31, 37	eqtr3id 2782	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
39		df-ov 7358	. . . . . 6 ⊢ (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩)
40	18, 7	eleqtrdi 2843	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
41	19, 8	eleqtrdi 2843	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))
42	34, 40, 41	swapf1 49433	. . . . . 6 ⊢ (𝜑 → (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ⟨𝑊, 𝑍⟩)
43	39, 42	eqtr3id 2782	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩) = ⟨𝑊, 𝑍⟩)
44	38, 43	oveq12d 7373	. . . 4 ⊢ (𝜑 → (((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩)) = (⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩))
45		df-ov 7358	. . . . 5 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
46	25	oveqi 7368	. . . . . . 7 ⊢ (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍)
47	22, 46	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑋(Hom ‘𝐶)𝑍))
48	26	oveqi 7368	. . . . . . 7 ⊢ (𝑌𝐽𝑊) = (𝑌(Hom ‘𝐷)𝑊)
49	23, 48	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑊))
50	34, 35, 36, 40, 41, 47, 49	swapf2 49435	. . . . 5 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ⟨𝑁, 𝑀⟩)
51	45, 50	eqtr3id 2782	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ⟨𝑁, 𝑀⟩)
52	44, 51	fveq12d 6838	. . 3 ⊢ (𝜑 → ((((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
53	4, 30, 52	3eqtrd 2772	. 2 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
54		df-ov 7358	. 2 ⊢ (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩)
55	53, 54	eqtr4di 2786	1 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4583   × cxp 5619  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Basecbs 17127  Hom chom 17179  Catccat 17578   Func cfunc 17769   ∘_func ccofu 17771   ×_c cxpc 18082   swap_F cswapf 49420

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

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-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  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 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-uz 12743  df-fz 13415  df-struct 17065  df-slot 17100  df-ndx 17112  df-base 17128  df-hom 17192  df-cco 17193  df-cat 17582  df-cid 17583  df-func 17773  df-cofu 17775  df-xpc 18086  df-swapf 49421

This theorem is referenced by:  tposcurf12  49459  tposcurf2  49461