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Mirrors > Home > MPE Home > Th. List > natcl | Structured version Visualization version GIF version |
Description: A component of a natural transformation is a morphism. (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) |
natcl.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
natcl | ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | natrcl.1 | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
2 | natixp.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) | |
3 | natixp.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
4 | natixp.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
5 | 1, 2, 3, 4 | natixp 17899 | . 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
6 | natcl.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | fveq2 6888 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
8 | fveq2 6888 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘𝑥) = (𝐾‘𝑋)) | |
9 | 7, 8 | oveq12d 7423 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) = ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
10 | 9 | fvixp 8892 | . 2 ⊢ ((𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
11 | 5, 6, 10 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟨cop 4633 ‘cfv 6540 (class class class)co 7405 Xcixp 8887 Basecbs 17140 Hom chom 17204 Nat cnat 17888 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-ixp 8888 df-func 17804 df-nat 17890 |
This theorem is referenced by: fuccocl 17913 fuclid 17915 fucrid 17916 fucass 17917 fucsect 17921 invfuc 17923 fucpropd 17926 evlfcllem 18170 evlfcl 18171 curfuncf 18187 yonedalem3a 18223 yonedalem3b 18228 yonedainv 18230 yonffthlem 18231 |
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