Theorem 1stfpropd 50519

Description: If two categories have the same set of objects, morphisms, and compositions, then they have same first 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
1stfpropd	⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷))

Proof of Theorem 1stfpropd

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 18389	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
10	9	fveq2d 6892	. . . 4 ⊢ (𝜑 → (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐵 ×c 𝐷)))
11	10	reseq2d 5970	. . 3 ⊢ (𝜑 → (1st ↾ (Base‘(𝐴 ×c 𝐶))) = (1st ↾ (Base‘(𝐵 ×c 𝐷))))
12	9	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐵 ×c 𝐷)))
13	12	oveqd 7439	. . . . 5 ⊢ (𝜑 → (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))
14	13	reseq2d 5970	. . . 4 ⊢ (𝜑 → (1st ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)) = (1st ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))
15	10, 10, 14	mpoeq123dv 7497	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (1st ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦))) = (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (1st ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))))
16	11, 15	opeq12d 4847	. 2 ⊢ (𝜑 → ⟨(1st ↾ (Base‘(𝐴 ×c 𝐶))), (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (1st ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)))⟩ = ⟨(1st ↾ (Base‘(𝐵 ×c 𝐷))), (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (1st ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))⟩)
17		eqid 2760	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
18		eqid 2760	. . 3 ⊢ (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐴 ×c 𝐶))
19		eqid 2760	. . 3 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
20		eqid 2760	. . 3 ⊢ (𝐴 1stF 𝐶) = (𝐴 1stF 𝐶)
21	17, 18, 19, 5, 7, 20	1stfval 18372	. 2 ⊢ (𝜑 → (𝐴 1stF 𝐶) = ⟨(1st ↾ (Base‘(𝐴 ×c 𝐶))), (𝑥 ∈ (Base‘(𝐴 ×c 𝐶)), 𝑦 ∈ (Base‘(𝐴 ×c 𝐶)) ↦ (1st ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)))⟩)
22		eqid 2760	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
23		eqid 2760	. . 3 ⊢ (Base‘(𝐵 ×c 𝐷)) = (Base‘(𝐵 ×c 𝐷))
24		eqid 2760	. . 3 ⊢ (Hom ‘(𝐵 ×c 𝐷)) = (Hom ‘(𝐵 ×c 𝐷))
25		eqid 2760	. . 3 ⊢ (𝐵 1stF 𝐷) = (𝐵 1stF 𝐷)
26	22, 23, 24, 6, 8, 25	1stfval 18372	. 2 ⊢ (𝜑 → (𝐵 1stF 𝐷) = ⟨(1st ↾ (Base‘(𝐵 ×c 𝐷))), (𝑥 ∈ (Base‘(𝐵 ×c 𝐷)), 𝑦 ∈ (Base‘(𝐵 ×c 𝐷)) ↦ (1st ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))⟩)
27	16, 21, 26	3eqtr4d 2805	1 ⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ⟨cop 4834   ↾ cres 5652  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424  1^st c1st 8001  Basecbs 17394  Hom chom 17446  Catccat 17845  Hom_f chomf 17847  comp_fccomf 17848   ×_c cxpc 18349   1^st_F c1stf 18350

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-uz 12971  df-fz 13645  df-struct 17332  df-slot 17367  df-ndx 17379  df-base 17395  df-hom 17459  df-cco 17460  df-homf 17851  df-comf 17852  df-xpc 18353  df-1stf 18354

This theorem is used by:  diagpropd  50521