Theorem 2ndfpropd 49262

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 18175	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
10	9	fveq2d 6864	. . . 4 ⊢ (𝜑 → (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐵 ×c 𝐷)))
11	10	reseq2d 5952	. . 3 ⊢ (𝜑 → (2nd ↾ (Base‘(𝐴 ×c 𝐶))) = (2nd ↾ (Base‘(𝐵 ×c 𝐷))))
12	9	fveq2d 6864	. . . . . 6 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐵 ×c 𝐷)))
13	12	oveqd 7406	. . . . 5 ⊢ (𝜑 → (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))
14	13	reseq2d 5952	. . . 4 ⊢ (𝜑 → (2nd ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)) = (2nd ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))
15	10, 10, 14	mpoeq123dv 7466	. . 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 4847	. 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 2730	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
18		eqid 2730	. . 3 ⊢ (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐴 ×c 𝐶))
19		eqid 2730	. . 3 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
20		eqid 2730	. . 3 ⊢ (𝐴 2ndF 𝐶) = (𝐴 2ndF 𝐶)
21	17, 18, 19, 5, 7, 20	2ndfval 18161	. 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 2730	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
23		eqid 2730	. . 3 ⊢ (Base‘(𝐵 ×c 𝐷)) = (Base‘(𝐵 ×c 𝐷))
24		eqid 2730	. . 3 ⊢ (Hom ‘(𝐵 ×c 𝐷)) = (Hom ‘(𝐵 ×c 𝐷))
25		eqid 2730	. . 3 ⊢ (𝐵 2ndF 𝐷) = (𝐵 2ndF 𝐷)
26	22, 23, 24, 6, 8, 25	2ndfval 18161	. 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 2775	1 ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  2^nd c2nd 7969  Basecbs 17185  Hom chom 17237  Catccat 17631  Hom_f chomf 17633  comp_fccomf 17634   ×_c cxpc 18135   2^nd_F c2ndf 18137

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-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-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  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-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-homf 17637  df-comf 17638  df-xpc 18139  df-2ndf 18141

This theorem is referenced by: (None)