Theorem uobrcl 49954

Description: Reverse closure for universal object. (Contributed by Zhi Wang, 17-Nov-2025.)

Assertion

Ref	Expression
uobrcl	⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Proof of Theorem uobrcl

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldmg 5890	. . . . 5 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ↔ ∃𝑚 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚))
2	1	ibi 270	. . . 4 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → ∃𝑚 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚)
3		simpr 489	. . . . . 6 ⊢ ((𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚) → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚)
4	3	up1st2nd 49946	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚) → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑚)
5	4	uprcl2 49950	. . . 4 ⊢ ((𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚) → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
6	2, 5	exlimddv 1965	. . 3 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
7	6	funcrcl2 49840	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐷 ∈ Cat)
8	6	funcrcl3 49841	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐸 ∈ Cat)
9	7, 8	jca 520	1 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1809   ∈ wcel 2143   class class class wbr 5110  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Catccat 17721   Func cfunc 17912   UP cup 49934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-func 17916  df-up 49935

This theorem is referenced by:  isinito4a  50309  initocmd  50430