Theorem cofuswapf2 49770

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 6844	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
3	2	oveqd 7384	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩))
4	3	oveqd 7384	. . 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 7370	. . . 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 2736	. . . . . 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 18144	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2736	. . . . . 6 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 49759	. . . . 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 5670	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18		cofuswapf2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
19		cofuswapf2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
20	18, 19	opelxpd 5670	. . . . 5 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
21		eqid 2736	. . . . 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 5670	. . . . . 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 18152	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
28	24, 27	eleqtrrd 2839	. . . . 5 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
29	9, 13, 14, 17, 20, 21, 28	cofu2 17853	. . . 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 2783	. . 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 7370	. . . . . 6 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
32	10, 11	swapfelvv 49738	. . . . . . . 8 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
33		1st2nd2 7981	. . . . . . . 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 2846	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
36	16, 8	eleqtrdi 2846	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
37	34, 35, 36	swapf1 49747	. . . . . 6 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
38	31, 37	eqtr3id 2785	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
39		df-ov 7370	. . . . . 6 ⊢ (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩)
40	18, 7	eleqtrdi 2846	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
41	19, 8	eleqtrdi 2846	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))
42	34, 40, 41	swapf1 49747	. . . . . 6 ⊢ (𝜑 → (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ⟨𝑊, 𝑍⟩)
43	39, 42	eqtr3id 2785	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩) = ⟨𝑊, 𝑍⟩)
44	38, 43	oveq12d 7385	. . . 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 7370	. . . . 5 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
46	25	oveqi 7380	. . . . . . 7 ⊢ (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍)
47	22, 46	eleqtrdi 2846	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑋(Hom ‘𝐶)𝑍))
48	26	oveqi 7380	. . . . . . 7 ⊢ (𝑌𝐽𝑊) = (𝑌(Hom ‘𝐷)𝑊)
49	23, 48	eleqtrdi 2846	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑊))
50	34, 35, 36, 40, 41, 47, 49	swapf2 49749	. . . . 5 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ⟨𝑁, 𝑀⟩)
51	45, 50	eqtr3id 2785	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ⟨𝑁, 𝑀⟩)
52	44, 51	fveq12d 6847	. . 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 2775	. 2 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
54		df-ov 7370	. 2 ⊢ (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩)
55	53, 54	eqtr4di 2789	1 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   × cxp 5629  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231  Catccat 17630   Func cfunc 17821   ∘_func ccofu 17823   ×_c cxpc 18134   swap_F cswapf 49734

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17634  df-cid 17635  df-func 17825  df-cofu 17827  df-xpc 18138  df-swapf 49735

This theorem is referenced by:  tposcurf12  49773  tposcurf2  49775