Theorem 2ndfpropd 50126

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 18286	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
10	9	fveq2d 6889	. . . 4 ⊢ (𝜑 → (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐵 ×c 𝐷)))
11	10	reseq2d 5980	. . 3 ⊢ (𝜑 → (2nd ↾ (Base‘(𝐴 ×c 𝐶))) = (2nd ↾ (Base‘(𝐵 ×c 𝐷))))
12	9	fveq2d 6889	. . . . . 6 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐵 ×c 𝐷)))
13	12	oveqd 7436	. . . . 5 ⊢ (𝜑 → (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))
14	13	reseq2d 5980	. . . 4 ⊢ (𝜑 → (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)) = (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))
15	10, 10, 14	mpoeq123dv 7494	. . 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 4848	. 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 2765	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
18		eqid 2765	. . 3 ⊢ (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐴 ×c 𝐶))
19		eqid 2765	. . 3 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
20		eqid 2765	. . 3 ⊢ (𝐴 2ndF 𝐶) = (𝐴 2ndF 𝐶)
21	17, 18, 19, 5, 7, 20	2ndfval 18272	. 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 2765	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
23		eqid 2765	. . 3 ⊢ (Base‘(𝐵 ×c 𝐷)) = (Base‘(𝐵 ×c 𝐷))
24		eqid 2765	. . 3 ⊢ (Hom ‘(𝐵 ×c 𝐷)) = (Hom ‘(𝐵 ×c 𝐷))
25		eqid 2765	. . 3 ⊢ (𝐵 2ndF 𝐷) = (𝐵 2ndF 𝐷)
26	22, 23, 24, 6, 8, 25	2ndfval 18272	. 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 2810	1 ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   ↾ cres 5665  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  2^nd c2nd 7991  Basecbs 17291  Hom chom 17343  Catccat 17742  Hom_f chomf 17744  comp_fccomf 17745   ×_c cxpc 18246   2^nd_F c2ndf 18248

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-struct 17229  df-slot 17264  df-ndx 17276  df-base 17292  df-hom 17356  df-cco 17357  df-homf 17748  df-comf 17749  df-xpc 18250  df-2ndf 18252

This theorem is used by: (None)