Theorem lmdpropd 49646

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 17864	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
10	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐴) = (Homf ‘𝐵))
11	10	oppchomfpropd 17687	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘𝐴)) = (Homf ‘(oppCat‘𝐵)))
12	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐴) = (compf‘𝐵))
13	10, 12	oppccomfpropd 17688	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘𝐴)) = (compf‘(oppCat‘𝐵)))
14	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘𝐶) = (compf‘𝐷))
16		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐶 Func 𝐴))
17	16	func1st2nd 49065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐶 Func 𝐴)(2nd ‘𝑓))
18	17	funcrcl2 49068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐶 ∈ Cat)
19	9	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 Func 𝐴) = (𝐷 Func 𝐵))
20	16, 19	eleqtrd 2830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐷 Func 𝐵))
21	20	func1st2nd 49065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐷 Func 𝐵)(2nd ‘𝑓))
22	21	funcrcl2 49068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐷 ∈ Cat)
23	17	funcrcl3 49069	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐴 ∈ Cat)
24	21	funcrcl3 49069	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐵 ∈ Cat)
25	14, 15, 10, 12, 18, 22, 23, 24	fucpropd 17942	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 FuncCat 𝐴) = (𝐷 FuncCat 𝐵))
26	25	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(𝐶 FuncCat 𝐴)) = (Homf ‘(𝐷 FuncCat 𝐵)))
27	26	oppchomfpropd 17687	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘(𝐶 FuncCat 𝐴))) = (Homf ‘(oppCat‘(𝐷 FuncCat 𝐵))))
28	25	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(𝐶 FuncCat 𝐴)) = (compf‘(𝐷 FuncCat 𝐵)))
29	26, 28	oppccomfpropd 17688	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘(𝐶 FuncCat 𝐴))) = (compf‘(oppCat‘(𝐷 FuncCat 𝐵))))
30		fvexd 6873	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐴) ∈ V)
31		fvexd 6873	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐵) ∈ V)
32		fvexd 6873	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐶 FuncCat 𝐴)) ∈ V)
33		fvexd 6873	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐷 FuncCat 𝐵)) ∈ V)
34	11, 13, 27, 29, 30, 31, 32, 33	uppropd 49170	. . . 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 49281	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
36	35	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → ( oppFunc ‘(𝐴Δfunc𝐶)) = ( oppFunc ‘(𝐵Δfunc𝐷)))
37		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 = 𝑓)
38	34, 36, 37	oveq123d 7408	. . 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 5193	. 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 49638	. 2 ⊢ (𝐴 Limit 𝐶) = (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓))
41		lmdfval 49638	. 2 ⊢ (𝐵 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐵) ↦ (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓))
42	39, 40, 41	3eqtr4g 2789	1 ⊢ (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Hom_f chomf 17627  comp_fccomf 17628  oppCatcoppc 17672   Func cfunc 17816   FuncCat cfuc 17907  Δ_funccdiag 18173   oppFunc coppf 49111   UP cup 49162   Limit clmd 49632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-homf 17631  df-comf 17632  df-oppc 17673  df-func 17820  df-nat 17908  df-fuc 17909  df-xpc 18133  df-1stf 18134  df-curf 18175  df-diag 18177  df-up 49163  df-lmd 49634

This theorem is referenced by:  termolmd  49659