Theorem oppc2ndf 49260

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 2730	. . . . . 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 8244	. . . . 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 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	7, 1	oppchom 17682	. . . . . . . . 9 ⊢ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦))
9		eqid 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	9, 2	oppchom 17682	. . . . . . . . 9 ⊢ ((2nd ‘𝑦)(Hom ‘𝑃)(2nd ‘𝑥)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))
11	8, 10	xpeq12i 5668	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝑂 ×c 𝑃) = (𝑂 ×c 𝑃)
13		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	1, 13	oppcbas 17685	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝑂)
15		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	2, 15	oppcbas 17685	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝑃)
17	12, 14, 16	xpcbas 18145	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝑂 ×c 𝑃))
18		eqid 2730	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
19		eqid 2730	. . . . . . . . 9 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
20		eqid 2730	. . . . . . . . 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 18147	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
25	24, 13, 15	xpcbas 18145	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
26		eqid 2730	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
27	24, 25, 7, 9, 26, 22, 21	xpchom 18147	. . . . . . . 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 2791	. . . . . . 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 5952	. . . . . 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 7468	. . . . 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 2784	. . . 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 4846	. . 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 2730	. . . . 5 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
36	24, 25, 26, 33, 34, 35	2ndfval 18161	. . . 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 18165	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
38	36, 37	oppfval3 49115	. . 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 17689	. . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
40	33, 39	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑂 ∈ Cat)
41	2	oppccat 17689	. . . . 5 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
42	34, 41	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑃 ∈ Cat)
43		eqid 2730	. . . 4 ⊢ (𝑂 2ndF 𝑃) = (𝑂 2ndF 𝑃)
44	12, 17, 20, 40, 42, 43	2ndfval 18161	. . 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 2775	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (oppFunc‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))
46		df-2ndf 18141	. 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 49258	1 ⊢ (𝜑 → (oppFunc‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⦋csb 3864  ⟨cop 4597   × cxp 5638   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206  Basecbs 17185  Hom chom 17237  Catccat 17631  oppCatcoppc 17678   ×_c cxpc 18135   2^nd_F c2ndf 18137  oppFunccoppf 49099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-homf 17637  df-comf 17638  df-oppc 17679  df-func 17826  df-xpc 18139  df-2ndf 18141  df-oppf 49100

This theorem is referenced by: (None)