Theorem uobrcl 49690

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 5847	. . . . 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 268	. . . 4 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → ∃𝑚 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚)
3		simpr 485	. . . . . 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 49682	. . . . 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 49686	. . . 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 1942	. . 3 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
7	6	funcrcl2 49576	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐷 ∈ Cat)
8	6	funcrcl3 49577	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐸 ∈ Cat)
9	7, 8	jca 516	1 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786   ∈ wcel 2119   class class class wbr 5079  dom cdm 5625  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  2^nd c2nd 7937  Catccat 17628   Func cfunc 17819   UP cup 49670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-func 17823  df-up 49671

This theorem is referenced by:  isinito4a  50045  initocmd  50166