Theorem lmdpropd 50319

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

Hypotheses

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

Assertion

Ref	Expression
lmdpropd	⊢ (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))

Proof of Theorem lmdpropd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmdpropd.3	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2		lmdpropd.4	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
3		lmdpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4		lmdpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
5		lmdpropd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
6		lmdpropd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
7		lmdpropd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		lmdpropd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 17958	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
10	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐴) = (Homf ‘𝐵))
11	10	oppchomfpropd 17781	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘𝐴)) = (Homf ‘(oppCat‘𝐵)))
12	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐴) = (compf‘𝐵))
13	10, 12	oppccomfpropd 17782	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘𝐴)) = (compf‘(oppCat‘𝐵)))
14	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐶) = (compf‘𝐷))
16		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐶 Func 𝐴))
17	16	func1st2nd 49738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐶 Func 𝐴)(2nd ‘𝑓))
18	17	funcrcl2 49741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐶 ∈ Cat)
19	9	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
20	16, 19	eleqtrd 2871	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐷 Func 𝐵))
21	20	func1st2nd 49738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐷 Func 𝐵)(2nd ‘𝑓))
22	21	funcrcl2 49741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐷 ∈ Cat)
23	17	funcrcl3 49742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐴 ∈ Cat)
24	21	funcrcl3 49742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐵 ∈ Cat)
25	14, 15, 10, 12, 18, 22, 23, 24	fucpropd 18036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 FuncCat 𝐴) = (𝐷 FuncCat 𝐵))
26	25	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(𝐶 FuncCat 𝐴)) = (Homf ‘(𝐷 FuncCat 𝐵)))
27	26	oppchomfpropd 17781	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘(𝐶 FuncCat 𝐴))) = (Homf ‘(oppCat‘(𝐷 FuncCat 𝐵))))
28	25	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(𝐶 FuncCat 𝐴)) = (compf‘(𝐷 FuncCat 𝐵)))
29	26, 28	oppccomfpropd 17782	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘(𝐶 FuncCat 𝐴))) = (compf‘(oppCat‘(𝐷 FuncCat 𝐵))))
30		fvexd 6897	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐴) ∈ V)
31		fvexd 6897	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐵) ∈ V)
32		fvexd 6897	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐶 FuncCat 𝐴)) ∈ V)
33		fvexd 6897	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐷 FuncCat 𝐵)) ∈ V)
34	11, 13, 27, 29, 30, 31, 32, 33	uppropd 49843	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → ((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴))) = ((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵))))
35	10, 12, 14, 15, 23, 24, 18, 22	diagpropd 49954	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
36	35	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → ( oppFunc ‘(𝐴Δfunc𝐶)) = ( oppFunc ‘(𝐵Δfunc𝐷)))
37		eqidd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 = 𝑓)
38	34, 36, 37	oveq123d 7432	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓) = (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓))
39	9, 38	mpteq12dva 5201	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓)) = (𝑓 ∈ (𝐷 Func 𝐵) ↦ (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓)))
40		lmdfval 50311	. 2 ⊢ (𝐴 Limit 𝐶) = (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓))
41		lmdfval 50311	. 2 ⊢ (𝐵 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐵) ↦ (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓))
42	39, 40, 41	3eqtr4g 2829	1 ⊢ (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  Hom_f chomf 17721  comp_fccomf 17722  oppCatcoppc 17766   Func cfunc 17910   FuncCat cfuc 18001  Δ_funccdiag 18267   oppFunc coppf 49784   UP cup 49835   Limit clmd 50305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-cat 17723  df-cid 17724  df-homf 17725  df-comf 17726  df-oppc 17767  df-func 17914  df-nat 18002  df-fuc 18003  df-xpc 18227  df-1stf 18228  df-curf 18269  df-diag 18271  df-up 49836  df-lmd 50307

This theorem is referenced by:  termolmd  50332