uobrcl - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > uobrcl		Structured version Visualization version GIF version

Theorem uobrcl 49100

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 5870	. . . . 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 49092	. . . . 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 49096	. . . 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 48996	. 2 ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → 𝐷 ∈ Cat)
8	6	funcrcl3 48997	. 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 5115 dom cdm 5646 ‘cfv 6519 (class class class)co 7394 1^st c1st 7975 2^nd c2nd 7976 Catccat 17631 Func cfunc 17822 UP cup 49081

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 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-ral 3047 df-rex 3056 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7977 df-2nd 7978 df-func 17826 df-up 49082

This theorem is referenced by: isinito4a 49426 initocmd 49538

W3C validator