Theorem ranval 50283

Description: Value of the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
lanval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
lanval.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
lanval.x	⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
ranval.k	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)
ranval.o	⊢ 𝑂 = (oppCat‘𝑅)
ranval.p	⊢ 𝑃 = (oppCat‘𝑆)

Assertion

Ref	Expression
ranval	⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))

Proof of Theorem ranval

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lanval.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
2		lanval.s	. . 3 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
3		lanval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4	3	func1st2nd 49739	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	4	funcrcl2 49742	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	4	funcrcl3 49743	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		lanval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
8	7	func1st2nd 49739	. . . 4 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
9	8	funcrcl3 49743	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
10		ranval.o	. . 3 ⊢ 𝑂 = (oppCat‘𝑅)
11		ranval.p	. . 3 ⊢ 𝑃 = (oppCat‘𝑆)
12	1, 2, 5, 6, 9, 10, 11	ranfval 50277	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂 UP 𝑃)𝑥)))
13		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
14	13	oveq2d 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
15		ranval.k	. . . . . . 7 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)
16	15	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)
17	14, 16	eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (⟨𝐷, 𝐸⟩ −∘F 𝑓) = ⟨𝐽, 𝐾⟩)
18	17	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓)) = ( oppFunc ‘⟨𝐽, 𝐾⟩))
19		df-ov 7414	. . . . . 6 ⊢ (𝐽 oppFunc 𝐾) = ( oppFunc ‘⟨𝐽, 𝐾⟩)
20	1, 9, 2, 3, 15	prcoffunca2 50050	. . . . . . 7 ⊢ (𝜑 → 𝐽(𝑅 Func 𝑆)𝐾)
21		oppfval 49799	. . . . . . 7 ⊢ (𝐽(𝑅 Func 𝑆)𝐾 → (𝐽 oppFunc 𝐾) = ⟨𝐽, tpos 𝐾⟩)
22	20, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐽 oppFunc 𝐾) = ⟨𝐽, tpos 𝐾⟩)
23	19, 22	eqtr3id 2818	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘⟨𝐽, 𝐾⟩) = ⟨𝐽, tpos 𝐾⟩)
24	23	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ( oppFunc ‘⟨𝐽, 𝐾⟩) = ⟨𝐽, tpos 𝐾⟩)
25	18, 24	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓)) = ⟨𝐽, tpos 𝐾⟩)
26		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
27	25, 26	oveq12d 7429	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂 UP 𝑃)𝑥) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
28		ovexd 7446	. 2 ⊢ (𝜑 → (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋) ∈ V)
29	12, 27, 3, 7, 28	ovmpod 7563	1 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8221  Catccat 17720  oppCatcoppc 17767   Func cfunc 17911   FuncCat cfuc 18002   oppFunc coppf 49785   UP cup 49836   −∘_F cprcof 50036   Ran cran 50269

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 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

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 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-cat 17724  df-cid 17725  df-func 17915  df-cofu 17917  df-nat 18003  df-fuc 18004  df-xpc 18228  df-curf 18270  df-oppf 49786  df-swapf 49923  df-fuco 49980  df-prcof 50037  df-ran 50271

This theorem is referenced by:  relran  50287  isran  50291  ranval2  50293