uobrcl - Mathbox for Zhi Wang

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Theorem uobrcl 49179

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 5845	. . . . 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 267	. . . 4 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → ∃𝑚 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚)
3		simpr 484	. . . . . 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 49171	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚) → 𝑋(⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑚)
5	4	uprcl2 49175	. . . 4 ⊢ ((𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) ∧ 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑚) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
6	2, 5	exlimddv 1935	. . 3 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
7	6	funcrcl2 49065	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐷 ∈ Cat)
8	6	funcrcl3 49066	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐸 ∈ Cat)
9	7, 8	jca 511	1 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2109 class class class wbr 5095 dom cdm 5623 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 Catccat 17588 Func cfunc 17779 UP cup 49159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-func 17783 df-up 49160

This theorem is referenced by: isinito4a 49534 initocmd 49655

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