fucoid2 - Mathbox for Zhi Wang

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Theorem fucoid2 49824

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid 49823. (Contributed by Zhi Wang, 30-Sep-2025.)

**Hypotheses**
Ref	Expression
fucoid.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucoid.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucoid.1	⊢ 1 = (Id‘𝑇)
fucoid.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoid.i	⊢ 𝐼 = (Id‘𝑄)
fucoid2.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
fucoid2.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)

**Assertion**
Ref	Expression
fucoid2	⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

**Proof of Theorem fucoid2**
Step	Hyp	Ref	Expression
1		fucoid.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fucoid.t	. 2 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3		fucoid.1	. 2 ⊢ 1 = (Id‘𝑇)
4		fucoid.q	. 2 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
5		fucoid.i	. 2 ⊢ 𝐼 = (Id‘𝑄)
6		fucoid2.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
7		fucoid2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8		relfunc 17829	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9		relfunc 17829	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
10	7, 6, 8, 9	fuco2eld2 49789	. . . . 5 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
11	6, 10, 7	3eltr3d 2850	. . . 4 ⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12		opelxp2 5674	. . . 4 ⊢ (⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ (𝐶 Func 𝐷))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ (𝐶 Func 𝐷))
14		df-br 5086	. . 3 ⊢ ((1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)) ↔ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ (𝐶 Func 𝐷))
15	13, 14	sylibr 234	. 2 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)))
16		opelxp1 5673	. . . 4 ⊢ (⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ (𝐷 Func 𝐸))
17	11, 16	syl 17	. . 3 ⊢ (𝜑 → ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ (𝐷 Func 𝐸))
18		df-br 5086	. . 3 ⊢ ((1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)) ↔ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ (𝐷 Func 𝐸))
19	17, 18	sylibr 234	. 2 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)))
20	1, 2, 3, 4, 5, 15, 19, 10	fucoid 49823	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4573 class class class wbr 5085 × cxp 5629 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 Idccid 17631 Func cfunc 17821 FuncCat cfuc 17912 ×_c cxpc 18134 ∘_F cfuco 49791

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17634 df-cid 17635 df-func 17825 df-cofu 17827 df-nat 17913 df-fuc 17914 df-xpc 18138 df-fuco 49792

This theorem is referenced by: fucofunc 49834

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