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| Description: A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| natrcl.1 | ⊢ 𝑁 = (𝐶 Nat 𝐷) | 
| natixp.2 | ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉)) | 
| natixp.b | ⊢ 𝐵 = (Base‘𝐶) | 
| natixp.j | ⊢ 𝐽 = (Hom ‘𝐷) | 
| Ref | Expression | 
|---|---|
| natixp | ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | natixp.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉)) | |
| 2 | natrcl.1 | . . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
| 3 | natixp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 4 | eqid 2736 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 5 | natixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐷) | |
| 6 | eqid 2736 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
| 7 | 2 | natrcl 17999 | . . . . . . 7 ⊢ (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) | 
| 8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) | 
| 9 | 8 | simpld 494 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | 
| 10 | df-br 5143 | . . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
| 11 | 9, 10 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) | 
| 12 | 8 | simprd 495 | . . . . 5 ⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) | 
| 13 | df-br 5143 | . . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) | |
| 14 | 12, 13 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) | 
| 15 | 2, 3, 4, 5, 6, 11, 14 | isnat 17996 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐷)(𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉(comp‘𝐷)(𝐾‘𝑦))(𝐴‘𝑥))))) | 
| 16 | 1, 15 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐷)(𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉(comp‘𝐷)(𝐾‘𝑦))(𝐴‘𝑥)))) | 
| 17 | 16 | simpld 494 | 1 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 〈cop 4631 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Xcixp 8938 Basecbs 17248 Hom chom 17309 compcco 17310 Func cfunc 17900 Nat cnat 17990 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-ixp 8939 df-func 17904 df-nat 17992 | 
| This theorem is referenced by: natcl 18002 natfn 18003 | 
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