Theorem ranval3 50959

Description: The set of right Kan extensions is the set of universal pairs. (Contributed by Zhi Wang, 26-Nov-2025.)

Hypotheses

Ref	Expression
ranval3.o	⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))
ranval3.p	⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))
ranval3.k	⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)

Assertion

Ref	Expression
ranval3	⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

Proof of Theorem ranval3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ranval3.o	. . 3 ⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))
2		ranval3.p	. . 3 ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))
3		id 23	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 ∈ (𝐶 Func 𝐷))
4		opex 5436	. . . . . 6 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ⟨𝐷, 𝐸⟩ ∈ V)
6	3, 5	prcofelvv 50660	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
7		1st2nd2 8050	. . . 4 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
8	6, 7	syl 18	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
9	1, 2, 8, 3	ranval2 50958	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
10		eqid 2763	. . . . . . . . 9 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
11	10	fucbas 18153	. . . . . . . 8 ⊢ (𝐶 Func 𝐸) = (Base‘(𝐶 FuncCat 𝐸))
12	2, 11	oppcbas 17907	. . . . . . 7 ⊢ (𝐶 Func 𝐸) = (Base‘𝑃)
13	12	uprcl 50445	. . . . . 6 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → (( oppFunc ‘𝐾) ∈ (𝑂 Func 𝑃) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
14	13	simprd 501	. . . . 5 ⊢ (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
15	14	adantl 487	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
16	12	uprcl 50445	. . . . . 6 ⊢ (𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋) → (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩ ∈ (𝑂 Func 𝑃) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
17	16	simprd 501	. . . . 5 ⊢ (𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋) → 𝑋 ∈ (𝐶 Func 𝐸))
18	17	adantl 487	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)) → 𝑋 ∈ (𝐶 Func 𝐸))
19		eqid 2763	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
20		funcrcl 18053	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
21	20	simprd 501	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐶 Func 𝐸) → 𝐸 ∈ Cat)
22	21	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐸 ∈ Cat)
23		simpl 488	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
24	19, 22, 10, 23	prcoffunca 50666	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)))
25		ranval3.k	. . . . . . . . 9 ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)
26	25	fveq2i 6899	. . . . . . . 8 ⊢ ( oppFunc ‘𝐾) = ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))
27		oppfval2 50398	. . . . . . . 8 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)) → ( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
28	26, 27	eqtrid 2810	. . . . . . 7 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func (𝐶 FuncCat 𝐸)) → ( oppFunc ‘𝐾) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
29	24, 28	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → ( oppFunc ‘𝐾) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
30	29	oveq1d 7445	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
31	30	eleq2d 2850	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)) → (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) ↔ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)))
32	15, 18, 31	bibiad 853	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝑥 ∈ (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) ↔ 𝑥 ∈ (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋)))
33	32	eqrdv 2761	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩(𝑂 UP 𝑃)𝑋))
34	9, 33	eqtr4d 2801	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ⟨cop 4838   × cxp 5653  ‘cfv 6547  (class class class)co 7430  1^st c1st 8009  2^nd c2nd 8010  tpos ctpos 8256  Catccat 17853  oppCatcoppc 17900   Func cfunc 18044   FuncCat cfuc 18135   oppFunc coppf 50383   UP cup 50434   −∘_F cprcof 50653   Ran cran 50934

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

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 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-tpos 8257  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-map 8863  df-ixp 8940  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-struct 17340  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-hom 17467  df-cco 17468  df-cat 17857  df-cid 17858  df-oppc 17901  df-func 18048  df-cofu 18050  df-nat 18136  df-fuc 18137  df-xpc 18361  df-curf 18403  df-oppf 50384  df-up 50435  df-swapf 50527  df-fuco 50584  df-prcof 50654  df-ran 50936

This theorem is used by:  lmdran  50999