fucoid2 - Mathbox for Zhi Wang

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Theorem fucoid2 49702

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid 49701. (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 17798	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9		relfunc 17798	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
10	7, 6, 8, 9	fuco2eld2 49667	. . . . 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 2851	. . . 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 5675	. . . 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 5101	. . 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 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 ‘(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 5101	. . 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 49701	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 class class class wbr 5100 × cxp 5630 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 Idccid 17600 Func cfunc 17790 FuncCat cfuc 17881 ×_c cxpc 18103 ∘_F cfuco 49669

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-cat 17603 df-cid 17604 df-func 17794 df-cofu 17796 df-nat 17882 df-fuc 17883 df-xpc 18107 df-fuco 49670

This theorem is referenced by: fucofunc 49712

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