uobeq3 - Mathbox for Zhi Wang

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Theorem uobeq3 50454

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 2761	. . . 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 50452	. . 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 50270	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 2145 ∩ cin 3898 dom cdm 5651 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7412 1^st c1st 7988 Basecbs 17367 Isociso 17901 Func cfunc 18009 ∘_func ccofu 18011 Full cful 18059 Faith cfth 18060 CatCatccatc 18253 UP cup 50225

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-slot 17340 df-ndx 17352 df-base 17368 df-hom 17432 df-cco 17433 df-cat 17822 df-cid 17823 df-sect 17902 df-inv 17903 df-iso 17904 df-func 18013 df-idfu 18014 df-cofu 18015 df-full 18061 df-fth 18062 df-catc 18254 df-up 50226

This theorem is used by: uobeqterm 50598

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