uobeq3 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > uobeq3		Structured version Visualization version GIF version

Theorem uobeq3 50221

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**
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	. . . 4 ⊢ 𝑄 = (CatCat‘𝑈)
7		eqid 2766	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
8		uobeq3.i	. . . 4 ⊢ 𝐼 = (Iso‘𝑄)
9		uobeq3.1	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝐷𝐼𝐸))
10	6, 1, 7, 8, 9	catcisoi 50219	. . 3 ⊢ (𝜑 → (𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)) ∧ (1^st ‘𝐾):𝐵–1-1-onto→(Base‘𝐸)))
11	10	simpld 500	. 2 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
12	1, 2, 3, 4, 5, 11	uobffth 50037	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ∩ cin 3907 dom cdm 5666 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7423 1^st c1st 7993 Basecbs 17294 Isociso 17828 Func cfunc 17936 ∘_func ccofu 17938 Full cful 17986 Faith cfth 17987 CatCatccatc 18180 UP cup 49992

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17232 df-slot 17267 df-ndx 17279 df-base 17295 df-hom 17359 df-cco 17360 df-cat 17749 df-cid 17750 df-sect 17829 df-inv 17830 df-iso 17831 df-func 17940 df-idfu 17941 df-cofu 17942 df-full 17988 df-fth 17989 df-catc 18181 df-up 49993

This theorem is used by: uobeqterm 50365

W3C validator