uobeq3 - Mathbox for Zhi Wang

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Theorem uobeq3 49294

Description: An isomorphism between categories generates equal sets of universal objects. (Contributed by Zhi Wang, 17-Nov-2025.)

**Hypotheses**
Ref	Expression
uobeq2.b	⊢ 𝐵 = (Base‘𝐷)
uobeq2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uobeq2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uobeq2.g	⊢ (𝜑 → (𝐾 ∘_func 𝐹) = 𝐺)
uobeq2.y	⊢ (𝜑 → ((1^st ‘𝐾)‘𝑋) = 𝑌)
uobeq2.q	⊢ 𝑄 = (CatCat‘𝑈)
uobeq3.i	⊢ 𝐼 = (Iso‘𝑄)
uobeq3.1	⊢ (𝜑 → 𝐾 ∈ (𝐷𝐼𝐸))

**Assertion**
Ref	Expression
uobeq3	⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Proof of Theorem uobeq3 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uobeq2.b	. 2 ⊢ 𝐵 = (Base‘𝐷)
2		uobeq2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		uobeq2.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		uobeq2.g	. 2 ⊢ (𝜑 → (𝐾 ∘_func 𝐹) = 𝐺)
5		uobeq2.y	. 2 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑋) = 𝑌)
6		uobeq2.q	. 2 ⊢ 𝑄 = (CatCat‘𝑈)
7		eqid 2730	. 2 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
8		eqid 2730	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		uobeq3.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
10		uobeq3.1	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷𝐼𝐸))
11	6, 1, 8, 9, 10	catcisoi 49292	. . . 4 ⊢ (𝜑 → (𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ (1^st ‘𝐾):𝐵–1-1-onto→(Base‘𝐸)))
12	11	simpld 494	. . 3 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
13	12	elin1d 4175	. 2 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
14		eqid 2730	. . . . . 6 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
15	14, 9	isoval2 48952	. . . . 5 ⊢ (𝐷𝐼𝐸) = dom (𝐷(Inv‘𝑄)𝐸)
16	10, 15	eleqtrdi 2839	. . . 4 ⊢ (𝜑 → 𝐾 ∈ dom (𝐷(Inv‘𝑄)𝐸))
17		eldmg 5870	. . . . 5 ⊢ (𝐾 ∈ dom (𝐷(Inv‘𝑄)𝐸) → (𝐾 ∈ dom (𝐷(Inv‘𝑄)𝐸) ↔ ∃𝑙 𝐾(𝐷(Inv‘𝑄)𝐸)𝑙))
18	17	ibi 267	. . . 4 ⊢ (𝐾 ∈ dom (𝐷(Inv‘𝑄)𝐸) → ∃𝑙 𝐾(𝐷(Inv‘𝑄)𝐸)𝑙)
19	14, 7	isinv2 48943	. . . . . 6 ⊢ (𝐾(𝐷(Inv‘𝑄)𝐸)𝑙 ↔ (𝐾(𝐷(Sect‘𝑄)𝐸)𝑙 ∧ 𝑙(𝐸(Sect‘𝑄)𝐷)𝐾))
20	19	simplbi 497	. . . . 5 ⊢ (𝐾(𝐷(Inv‘𝑄)𝐸)𝑙 → 𝐾(𝐷(Sect‘𝑄)𝐸)𝑙)
21	20	eximi 1835	. . . 4 ⊢ (∃𝑙 𝐾(𝐷(Inv‘𝑄)𝐸)𝑙 → ∃𝑙 𝐾(𝐷(Sect‘𝑄)𝐸)𝑙)
22	16, 18, 21	3syl 18	. . 3 ⊢ (𝜑 → ∃𝑙 𝐾(𝐷(Sect‘𝑄)𝐸)𝑙)
23		eldmg 5870	. . . 4 ⊢ (𝐾 ∈ (𝐷𝐼𝐸) → (𝐾 ∈ dom (𝐷(Sect‘𝑄)𝐸) ↔ ∃𝑙 𝐾(𝐷(Sect‘𝑄)𝐸)𝑙))
24	10, 23	syl 17	. . 3 ⊢ (𝜑 → (𝐾 ∈ dom (𝐷(Sect‘𝑄)𝐸) ↔ ∃𝑙 𝐾(𝐷(Sect‘𝑄)𝐸)𝑙))
25	22, 24	mpbird 257	. 2 ⊢ (𝜑 → 𝐾 ∈ dom (𝐷(Sect‘𝑄)𝐸))
26	1, 2, 3, 4, 5, 6, 7, 13, 25	uobeq2 49293	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∩ cin 3921 class class class wbr 5115 dom cdm 5646 –1-1-onto→wf1o 6518 ‘cfv 6519 (class class class)co 7394 1^st c1st 7975 Basecbs 17185 Sectcsect 17712 Invcinv 17713 Isociso 17714 Func cfunc 17822 ∘_func ccofu 17824 Full cful 17872 Faith cfth 17873 CatCatccatc 18066 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 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 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-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-we 5601 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-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 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-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7851 df-1st 7977 df-2nd 7978 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-er 8682 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-z 12546 df-dec 12666 df-uz 12810 df-fz 13482 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-cat 17635 df-cid 17636 df-sect 17715 df-inv 17716 df-iso 17717 df-func 17826 df-idfu 17827 df-cofu 17828 df-full 17874 df-fth 17875 df-catc 18067 df-up 49082

This theorem is referenced by: uobeqterm 49424

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