Theorem oppc2ndf 49534

Description: The opposite functor of the second projection functor is the second projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)

Hypotheses

Ref	Expression
oppc1stf.o	⊢ 𝑂 = (oppCat‘𝐶)
oppc1stf.p	⊢ 𝑃 = (oppCat‘𝐷)
oppc1stf.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
oppc1stf.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
oppc2ndf	⊢ (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))

Proof of Theorem oppc2ndf

Dummy variables 𝑥 𝑦 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppc1stf.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
2		oppc1stf.p	. 2 ⊢ 𝑃 = (oppCat‘𝐷)
3		oppc1stf.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		oppc1stf.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))
6	5	tposmpo 8205	. . . . 5 ⊢ tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))
7		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	7, 1	oppchom 17638	. . . . . . . . 9 ⊢ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦))
9		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	9, 2	oppchom 17638	. . . . . . . . 9 ⊢ ((2nd ‘𝑦)(Hom ‘𝑃)(2nd ‘𝑥)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))
11	8, 10	xpeq12i 5652	. . . . . . . 8 ⊢ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑥)) × ((2nd ‘𝑦)(Hom ‘𝑃)(2nd ‘𝑥))) = (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))
12		eqid 2736	. . . . . . . . 9 ⊢ (𝑂 ×c 𝑃) = (𝑂 ×c 𝑃)
13		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	1, 13	oppcbas 17641	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝑂)
15		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	2, 15	oppcbas 17641	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝑃)
17	12, 14, 16	xpcbas 18101	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝑂 ×c 𝑃))
18		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
19		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
20		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘(𝑂 ×c 𝑃)) = (Hom ‘(𝑂 ×c 𝑃))
21		simp2 1137	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
22		simp3 1138	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
23	12, 17, 18, 19, 20, 21, 22	xpchom 18103	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥) = (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑥)) × ((2nd ‘𝑦)(Hom ‘𝑃)(2nd ‘𝑥))))
24		eqid 2736	. . . . . . . . 9 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
25	24, 13, 15	xpcbas 18101	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
26		eqid 2736	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
27	24, 25, 7, 9, 26, 22, 21	xpchom 18103	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦) = (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))
28	11, 23, 27	3eqtr4a 2797	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥) = (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦))
29	28	reseq2d 5938	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd ↾ (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥)) = (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))
30	29	mpoeq3dva 7435	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦))))
31	6, 30	eqtr4id 2790	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥))))
32	31	opeq2d 4836	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))⟩ = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥)))⟩)
33		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
34		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐷 ∈ Cat)
35		eqid 2736	. . . . 5 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
36	24, 25, 26, 33, 34, 35	2ndfval 18117	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 2ndF 𝐷) = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))⟩)
37	24, 33, 34, 35	2ndfcl 18121	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
38	36, 37	oppfval3 49383	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 2ndF 𝐷)) = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐶 ×c 𝐷))𝑦)))⟩)
39	1	oppccat 17645	. . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
40	33, 39	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑂 ∈ Cat)
41	2	oppccat 17645	. . . . 5 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
42	34, 41	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑃 ∈ Cat)
43		eqid 2736	. . . 4 ⊢ (𝑂 2ndF 𝑃) = (𝑂 2ndF 𝑃)
44	12, 17, 20, 40, 42, 43	2ndfval 18117	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑂 2ndF 𝑃) = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑦(Hom ‘(𝑂 ×c 𝑃))𝑥)))⟩)
45	32, 38, 44	3eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))
46		df-2ndf 18097	. 2 ⊢ 2ndF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑐 ×c 𝑑))𝑦)))⟩)
47	1, 2, 3, 4, 45, 46	oppc1stflem 49532	1 ⊢ (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⦋csb 3849  ⟨cop 4586   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  tpos ctpos 8167  Basecbs 17136  Hom chom 17188  Catccat 17587  oppCatcoppc 17634   ×_c cxpc 18091   2^nd_F c2ndf 18093   oppFunc coppf 49367

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-homf 17593  df-comf 17594  df-oppc 17635  df-func 17782  df-xpc 18095  df-2ndf 18097  df-oppf 49368

This theorem is referenced by: (None)