Theorem uobrcl 49778

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 5872	. . . . 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 269	. . . 4 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → ∃𝑚 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚)
3		simpr 488	. . . . . 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 49770	. . . . 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 49774	. . . 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 1954	. . 3 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
7	6	funcrcl2 49664	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐷 ∈ Cat)
8	6	funcrcl3 49665	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐸 ∈ Cat)
9	7, 8	jca 519	1 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1798   ∈ wcel 2141   class class class wbr 5099  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965  Catccat 17679   Func cfunc 17870   UP cup 49758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-func 17874  df-up 49759

This theorem is referenced by:  isinito4a  50133  initocmd  50254