Theorem lmdpropd 50144

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 17860	. . 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 17683	. . . . 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 17684	. . . . 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 49563	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐶 Func 𝐴)(2nd ‘𝑓))
18	17	funcrcl2 49566	. . . . . . . 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 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 ∈ (𝐷 Func 𝐵))
21	20	func1st2nd 49563	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (1st ‘𝑓)(𝐷 Func 𝐵)(2nd ‘𝑓))
22	21	funcrcl2 49566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐷 ∈ Cat)
23	17	funcrcl3 49567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐴 ∈ Cat)
24	21	funcrcl3 49567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝐵 ∈ Cat)
25	14, 15, 10, 12, 18, 22, 23, 24	fucpropd 17938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐶 FuncCat 𝐴) = (𝐷 FuncCat 𝐵))
26	25	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(𝐶 FuncCat 𝐴)) = (Homf ‘(𝐷 FuncCat 𝐵)))
27	26	oppchomfpropd 17683	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (Homf ‘(oppCat‘(𝐶 FuncCat 𝐴))) = (Homf ‘(oppCat‘(𝐷 FuncCat 𝐵))))
28	25	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(𝐶 FuncCat 𝐴)) = (compf‘(𝐷 FuncCat 𝐵)))
29	26, 28	oppccomfpropd 17684	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (compf‘(oppCat‘(𝐶 FuncCat 𝐴))) = (compf‘(oppCat‘(𝐷 FuncCat 𝐵))))
30		fvexd 6849	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐴) ∈ V)
31		fvexd 6849	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘𝐵) ∈ V)
32		fvexd 6849	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐶 FuncCat 𝐴)) ∈ V)
33		fvexd 6849	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (oppCat‘(𝐷 FuncCat 𝐵)) ∈ V)
34	11, 13, 27, 29, 30, 31, 32, 33	uppropd 49668	. . . 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 49779	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
36	35	fveq2d 6838	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → ( oppFunc ‘(𝐴Δfunc𝐶)) = ( oppFunc ‘(𝐵Δfunc𝐷)))
37		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐶 Func 𝐴)) → 𝑓 = 𝑓)
38	34, 36, 37	oveq123d 7381	. . 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 5172	. 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 50136	. 2 ⊢ (𝐴 Limit 𝐶) = (𝑓 ∈ (𝐶 Func 𝐴) ↦ (( oppFunc ‘(𝐴Δfunc𝐶))((oppCat‘𝐴) UP (oppCat‘(𝐶 FuncCat 𝐴)))𝑓))
41		lmdfval 50136	. 2 ⊢ (𝐵 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐵) ↦ (( oppFunc ‘(𝐵Δfunc𝐷))((oppCat‘𝐵) UP (oppCat‘(𝐷 FuncCat 𝐵)))𝑓))
42	39, 40, 41	3eqtr4g 2797	1 ⊢ (𝜑 → (𝐴 Limit 𝐶) = (𝐵 Limit 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Hom_f chomf 17623  comp_fccomf 17624  oppCatcoppc 17668   Func cfunc 17812   FuncCat cfuc 17903  Δ_funccdiag 18169   oppFunc coppf 49609   UP cup 49660   Limit clmd 50130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-oppc 17669  df-func 17816  df-nat 17904  df-fuc 17905  df-xpc 18129  df-1stf 18130  df-curf 18171  df-diag 18173  df-up 49661  df-lmd 50132

This theorem is referenced by:  termolmd  50157