Theorem cofuswapf2 49040

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 (𝐶swapF𝐷)))
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 (𝐶swapF𝐷)))
2	1	fveq2d 6890	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘(𝐹 ∘func (𝐶swapF𝐷))))
3	2	oveqd 7430	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩))
4	3	oveqd 7430	. . 3 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩)𝑁))
5		df-ov 7416	. . . 4 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
6		eqid 2734	. . . . . 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 18194	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2734	. . . . . 6 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 49035	. . . . 5 ⊢ (𝜑 → (𝐶swapF𝐷) ∈ ((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 5704	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18		cofuswapf2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
19		cofuswapf2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
20	18, 19	opelxpd 5704	. . . . 5 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
21		eqid 2734	. . . . 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 5704	. . . . . 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 18202	. . . . . 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 17903	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ((((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶swapF𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)))
30	5, 29	eqtrid 2781	. . 3 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶swapF𝐷)))⟨𝑍, 𝑊⟩)𝑁) = ((((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶swapF𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)))
31		df-ov 7416	. . . . . 6 ⊢ (𝑋(1st ‘(𝐶swapF𝐷))𝑌) = ((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩)
32	10, 11	swapfelvv 49014	. . . . . . . 8 ⊢ (𝜑 → (𝐶swapF𝐷) ∈ (V × V))
33		1st2nd2 8035	. . . . . . . 8 ⊢ ((𝐶swapF𝐷) ∈ (V × V) → (𝐶swapF𝐷) = ⟨(1st ‘(𝐶swapF𝐷)), (2nd ‘(𝐶swapF𝐷))⟩)
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨(1st ‘(𝐶swapF𝐷)), (2nd ‘(𝐶swapF𝐷))⟩)
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 49023	. . . . . 6 ⊢ (𝜑 → (𝑋(1st ‘(𝐶swapF𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
38	31, 37	eqtr3id 2783	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
39		df-ov 7416	. . . . . 6 ⊢ (𝑍(1st ‘(𝐶swapF𝐷))𝑊) = ((1st ‘(𝐶swapF𝐷))‘⟨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 49023	. . . . . 6 ⊢ (𝜑 → (𝑍(1st ‘(𝐶swapF𝐷))𝑊) = ⟨𝑊, 𝑍⟩)
43	39, 42	eqtr3id 2783	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶swapF𝐷))‘⟨𝑍, 𝑊⟩) = ⟨𝑊, 𝑍⟩)
44	38, 43	oveq12d 7431	. . . 4 ⊢ (𝜑 → (((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶swapF𝐷))‘⟨𝑍, 𝑊⟩)) = (⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩))
45		df-ov 7416	. . . . 5 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
46	25	oveqi 7426	. . . . . . 7 ⊢ (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍)
47	22, 46	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑋(Hom ‘𝐶)𝑍))
48	26	oveqi 7426	. . . . . . 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 49025	. . . . 5 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)𝑁) = ⟨𝑁, 𝑀⟩)
51	45, 50	eqtr3id 2783	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ⟨𝑁, 𝑀⟩)
52	44, 51	fveq12d 6893	. . 3 ⊢ (𝜑 → ((((1st ‘(𝐶swapF𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐹)((1st ‘(𝐶swapF𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶swapF𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
53	4, 30, 52	3eqtrd 2773	. 2 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
54		df-ov 7416	. 2 ⊢ (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩)
55	53, 54	eqtr4di 2787	1 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612   × cxp 5663  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  Basecbs 17230  Hom chom 17285  Catccat 17679   Func cfunc 17871   ∘_func ccofu 17873   ×_c cxpc 18184  swap_Fcswapf 49010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-xpc 18188  df-swapf 49011

This theorem is referenced by:  tposcurf12  49043  tposcurf2  49045