Theorem cofuswapf2 50402

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 6889	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) = (2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
3	2	oveqd 7437	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑍, 𝑊⟩))
4	3	oveqd 7437	. . 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 7423	. . . 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 2761	. . . . . 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 18352	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2761	. . . . . 6 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 50391	. . . . 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 5690	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18		cofuswapf2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
19		cofuswapf2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
20	18, 19	opelxpd 5690	. . . . 5 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
21		eqid 2761	. . . . 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 5690	. . . . . 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 18360	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
28	24, 27	eleqtrrd 2864	. . . . 5 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
29	9, 13, 14, 17, 20, 21, 28	cofu2 18061	. . . 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 2808	. . 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 7423	. . . . . 6 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
32	10, 11	swapfelvv 50370	. . . . . . . 8 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
33		1st2nd2 8040	. . . . . . . 8 ⊢ ((𝐶 swapF 𝐷) ∈ (V × V) → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
34	32, 33	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
35	15, 7	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
36	16, 8	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
37	34, 35, 36	swapf1 50379	. . . . . 6 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
38	31, 37	eqtr3id 2810	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
39		df-ov 7423	. . . . . 6 ⊢ (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩)
40	18, 7	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
41	19, 8	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))
42	34, 40, 41	swapf1 50379	. . . . . 6 ⊢ (𝜑 → (𝑍(1st ‘(𝐶 swapF 𝐷))𝑊) = ⟨𝑊, 𝑍⟩)
43	39, 42	eqtr3id 2810	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑍, 𝑊⟩) = ⟨𝑊, 𝑍⟩)
44	38, 43	oveq12d 7438	. . . 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 7423	. . . . 5 ⊢ (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩)
46	25	oveqi 7433	. . . . . . 7 ⊢ (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍)
47	22, 46	eleqtrdi 2871	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑋(Hom ‘𝐶)𝑍))
48	26	oveqi 7433	. . . . . . 7 ⊢ (𝑌𝐽𝑊) = (𝑌(Hom ‘𝐷)𝑊)
49	23, 48	eleqtrdi 2871	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑊))
50	34, 35, 36, 40, 41, 47, 49	swapf2 50381	. . . . 5 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)𝑁) = ⟨𝑁, 𝑀⟩)
51	45, 50	eqtr3id 2810	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 swapF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑀, 𝑁⟩) = ⟨𝑁, 𝑀⟩)
52	44, 51	fveq12d 6892	. . 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 2800	. 2 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩))
54		df-ov 7423	. 2 ⊢ (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀) = ((⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)‘⟨𝑁, 𝑀⟩)
55	53, 54	eqtr4di 2814	1 ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   × cxp 5649  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Hom chom 17439  Catccat 17838   Func cfunc 18029   ∘_func ccofu 18031   ×_c cxpc 18342   swap_F cswapf 50366

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 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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-rmo 3366  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035  df-xpc 18346  df-swapf 50367

This theorem is used by:  tposcurf12  50405  tposcurf2  50407