Theorem lmdpropd 50976

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 18086	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
10	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐴) = (Homf ‘𝐵))
11	10	oppchomfpropd 17909	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘𝐴)) = (Homf ‘(oppCat‘𝐵)))
12	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐴) = (compf‘𝐵))
13	10, 12	oppccomfpropd 17910	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘𝐴)) = (compf‘(oppCat‘𝐵)))
14	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐶) = (compf‘𝐷))
16		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐶 Func 𝐴))
17	16	func1st2nd 50326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐶 Func 𝐴)(2nd ‘𝑓))
18	17	funcrcl2 50329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐶 ∈ Cat)
19	9	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
20	16, 19	eleqtrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐷 Func 𝐵))
21	20	func1st2nd 50326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐷 Func 𝐵)(2nd ‘𝑓))
22	21	funcrcl2 50329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐷 ∈ Cat)
23	17	funcrcl3 50330	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐴 ∈ Cat)
24	21	funcrcl3 50330	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐵 ∈ Cat)
25	14, 15, 10, 12, 18, 22, 23, 24	fucpropd 18164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 FuncCat 𝐴) = (𝐷 FuncCat 𝐵))
26	25	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(𝐶 FuncCat 𝐴)) = (Homf ‘(𝐷 FuncCat 𝐵)))
27	26	oppchomfpropd 17909	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘(𝐶 FuncCat 𝐴))) = (Homf ‘(oppCat‘(𝐷 FuncCat 𝐵))))
28	25	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(𝐶 FuncCat 𝐴)) = (compf‘(𝐷 FuncCat 𝐵)))
29	26, 28	oppccomfpropd 17910	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘(𝐶 FuncCat 𝐴))) = (compf‘(oppCat‘(𝐷 FuncCat 𝐵))))
30		fvexd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐴) ∈ V)
31		fvexd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐵) ∈ V)
32		fvexd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐶 FuncCat 𝐴)) ∈ V)
33		fvexd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐷 FuncCat 𝐵)) ∈ V)
34	11, 13, 27, 29, 30, 31, 32, 33	uppropd 50433	. . . 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 50550	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
36	35	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → ( oppFunc ‘(𝐴Δfunc𝐶)) = ( oppFunc ‘(𝐵Δfunc𝐷)))
37		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 = 𝑓)
38	34, 36, 37	oveq123d 7445	. . 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 5190	. 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 50968	. 2 ⊢ (𝐴 Limit 𝐶) = (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓))
41		lmdfval 50968	. 2 ⊢ (𝐵 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐵) ↦ (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓))
42	39, 40, 41	3eqtr4g 2820	1 ⊢ (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004  Hom_f chomf 17849  comp_fccomf 17850  oppCatcoppc 17894   Func cfunc 18038   FuncCat cfuc 18129  Δ_funccdiag 18395   oppFunc coppf 50374   UP cup 50425   Limit clmd 50962

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 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

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-rmo 3365  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-tpos 8251  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-map 8857  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-struct 17334  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-hom 17461  df-cco 17462  df-cat 17851  df-cid 17852  df-homf 17853  df-comf 17854  df-oppc 17895  df-func 18042  df-nat 18130  df-fuc 18131  df-xpc 18355  df-1stf 18356  df-curf 18397  df-diag 18399  df-up 50426  df-lmd 50964

This theorem is used by:  termolmd  50989