Theorem lmdpropd 50584

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 17992	. . 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 17815	. . . . 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 17816	. . . . 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 50003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐶 Func 𝐴)(2nd ‘𝑓))
18	17	funcrcl2 50006	. . . . . . . 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 50003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐷 Func 𝐵)(2nd ‘𝑓))
22	21	funcrcl2 50006	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐷 ∈ Cat)
23	17	funcrcl3 50007	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐴 ∈ Cat)
24	21	funcrcl3 50007	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐵 ∈ Cat)
25	14, 15, 10, 12, 18, 22, 23, 24	fucpropd 18070	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 FuncCat 𝐴) = (𝐷 FuncCat 𝐵))
26	25	fveq2d 6883	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(𝐶 FuncCat 𝐴)) = (Homf ‘(𝐷 FuncCat 𝐵)))
27	26	oppchomfpropd 17815	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘(𝐶 FuncCat 𝐴))) = (Homf ‘(oppCat‘(𝐷 FuncCat 𝐵))))
28	25	fveq2d 6883	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(𝐶 FuncCat 𝐴)) = (compf‘(𝐷 FuncCat 𝐵)))
29	26, 28	oppccomfpropd 17816	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘(𝐶 FuncCat 𝐴))) = (compf‘(oppCat‘(𝐷 FuncCat 𝐵))))
30		fvexd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐴) ∈ V)
31		fvexd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐵) ∈ V)
32		fvexd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐶 FuncCat 𝐴)) ∈ V)
33		fvexd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐷 FuncCat 𝐵)) ∈ V)
34	11, 13, 27, 29, 30, 31, 32, 33	uppropd 50108	. . . 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 50219	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
36	35	fveq2d 6883	. . . 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 7435	. . 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 5191	. 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 50576	. 2 ⊢ (𝐴 Limit 𝐶) = (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓))
41		lmdfval 50576	. 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 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  Hom_f chomf 17755  comp_fccomf 17756  oppCatcoppc 17800   Func cfunc 17944   FuncCat cfuc 18035  Δ_funccdiag 18301   oppFunc coppf 50049   UP cup 50100   Limit clmd 50570

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

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 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-map 8829  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13563  df-struct 17240  df-sets 17257  df-slot 17275  df-ndx 17287  df-base 17303  df-hom 17367  df-cco 17368  df-cat 17757  df-cid 17758  df-homf 17759  df-comf 17760  df-oppc 17801  df-func 17948  df-nat 18036  df-fuc 18037  df-xpc 18261  df-1stf 18262  df-curf 18303  df-diag 18305  df-up 50101  df-lmd 50572

This theorem is used by:  termolmd  50597