Theorem 2ndfpropd 49452

Description: If two categories have the same set of objects, morphisms, and compositions, then they have same second projection functors. (Contributed by Zhi Wang, 20-Nov-2025.)

Hypotheses

Ref	Expression
1stfpropd.1	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
1stfpropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
1stfpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
1stfpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
1stfpropd.a	⊢ (𝜑 → 𝐴 ∈ Cat)
1stfpropd.b	⊢ (𝜑 → 𝐵 ∈ Cat)
1stfpropd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
1stfpropd.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
2ndfpropd	⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))

Proof of Theorem 2ndfpropd

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

Step	Hyp	Ref	Expression
1		1stfpropd.1	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		1stfpropd.2	. . . . . 6 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		1stfpropd.3	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		1stfpropd.4	. . . . . 6 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		1stfpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Cat)
6		1stfpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Cat)
7		1stfpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		1stfpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	1, 2, 3, 4, 5, 6, 7, 8	xpcpropd 18122	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
10	9	fveq2d 6835	. . . 4 ⊢ (𝜑 → (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐵 ×c 𝐷)))
11	10	reseq2d 5935	. . 3 ⊢ (𝜑 → (2nd ↾ (Base‘(𝐴 ×c 𝐶))) = (2nd ↾ (Base‘(𝐵 ×c 𝐷))))
12	9	fveq2d 6835	. . . . . 6 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐵 ×c 𝐷)))
13	12	oveqd 7372	. . . . 5 ⊢ (𝜑 → (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))
14	13	reseq2d 5935	. . . 4 ⊢ (𝜑 → (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)) = (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))
15	10, 10, 14	mpoeq123dv 7430	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦))) = (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))))
16	11, 15	opeq12d 4834	. 2 ⊢ (𝜑 → ⟨(2nd ↾ (Base‘(𝐴 ×c 𝐶))), (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)))⟩ = ⟨(2nd ↾ (Base‘(𝐵 ×c 𝐷))), (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))⟩)
17		eqid 2733	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
18		eqid 2733	. . 3 ⊢ (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐴 ×c 𝐶))
19		eqid 2733	. . 3 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
20		eqid 2733	. . 3 ⊢ (𝐴 2ndF 𝐶) = (𝐴 2ndF 𝐶)
21	17, 18, 19, 5, 7, 20	2ndfval 18108	. 2 ⊢ (𝜑 → (𝐴 2ndF 𝐶) = ⟨(2nd ↾ (Base‘(𝐴 ×c 𝐶))), (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)))⟩)
22		eqid 2733	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
23		eqid 2733	. . 3 ⊢ (Base‘(𝐵 ×c 𝐷)) = (Base‘(𝐵 ×c 𝐷))
24		eqid 2733	. . 3 ⊢ (Hom ‘(𝐵 ×c 𝐷)) = (Hom ‘(𝐵 ×c 𝐷))
25		eqid 2733	. . 3 ⊢ (𝐵 2ndF 𝐷) = (𝐵 2ndF 𝐷)
26	22, 23, 24, 6, 8, 25	2ndfval 18108	. 2 ⊢ (𝜑 → (𝐵 2ndF 𝐷) = ⟨(2nd ↾ (Base‘(𝐵 ×c 𝐷))), (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))⟩)
27	16, 21, 26	3eqtr4d 2778	1 ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   ↾ cres 5623  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  2^nd c2nd 7929  Basecbs 17127  Hom chom 17179  Catccat 17578  Hom_f chomf 17580  comp_fccomf 17581   ×_c cxpc 18082   2^nd_F c2ndf 18084

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 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-uz 12743  df-fz 13415  df-struct 17065  df-slot 17100  df-ndx 17112  df-base 17128  df-hom 17192  df-cco 17193  df-homf 17584  df-comf 17585  df-xpc 18086  df-2ndf 18088

This theorem is referenced by: (None)