Theorem 1stfpropd 50743

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 18432	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
10	9	fveq2d 6920	. . . 4 ⊢ (𝜑 → (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐵 ×c 𝐷)))
11	10	reseq2d 5986	. . 3 ⊢ (𝜑 → (1st ↾ (Base‘(𝐴 ×c 𝐶))) = (1st ↾ (Base‘(𝐵 ×c 𝐷))))
12	9	fveq2d 6920	. . . . . 6 ⊢ (𝜑 → (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐵 ×c 𝐷)))
13	12	oveqd 7468	. . . . 5 ⊢ (𝜑 → (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦))
14	13	reseq2d 5986	. . . 4 ⊢ (𝜑 → (1st ↾ (𝑥(Hom ‘(𝐴 ×c 𝐶))𝑦)) = (1st ↾ (𝑥(Hom ‘(𝐵 ×c 𝐷))𝑦)))
15	10, 10, 14	mpoeq123dv 7526	. . 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 4853	. 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 2763	. . 3 ⊢ (𝐴 ×c 𝐶) = (𝐴 ×c 𝐶)
18		eqid 2763	. . 3 ⊢ (Base‘(𝐴 ×c 𝐶)) = (Base‘(𝐴 ×c 𝐶))
19		eqid 2763	. . 3 ⊢ (Hom ‘(𝐴 ×c 𝐶)) = (Hom ‘(𝐴 ×c 𝐶))
20		eqid 2763	. . 3 ⊢ (𝐴 1stF 𝐶) = (𝐴 1stF 𝐶)
21	17, 18, 19, 5, 7, 20	1stfval 18415	. 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 2763	. . 3 ⊢ (𝐵 ×c 𝐷) = (𝐵 ×c 𝐷)
23		eqid 2763	. . 3 ⊢ (Base‘(𝐵 ×c 𝐷)) = (Base‘(𝐵 ×c 𝐷))
24		eqid 2763	. . 3 ⊢ (Hom ‘(𝐵 ×c 𝐷)) = (Hom ‘(𝐵 ×c 𝐷))
25		eqid 2763	. . 3 ⊢ (𝐵 1stF 𝐷) = (𝐵 1stF 𝐷)
26	22, 23, 24, 6, 8, 25	1stfval 18415	. 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 2808	1 ⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⟨cop 4840   ↾ cres 5660  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  Basecbs 17431  Hom chom 17483  Catccat 17888  Hom_f chomf 17890  comp_fccomf 17891   ×_c cxpc 18392   1^st_F c1stf 18393

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-homf 17894  df-comf 17895  df-xpc 18396  df-1stf 18397

This theorem is used by:  diagpropd  50745