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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 17900 | . 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
6 | natcl.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | fveq2 6889 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
8 | fveq2 6889 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘𝑥) = (𝐾‘𝑋)) | |
9 | 7, 8 | oveq12d 7424 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) = ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
10 | 9 | fvixp 8893 | . 2 ⊢ ((𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
11 | 5, 6, 10 | syl2anc 585 | 1 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 〈cop 4634 ‘cfv 6541 (class class class)co 7406 Xcixp 8888 Basecbs 17141 Hom chom 17205 Nat cnat 17889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-ixp 8889 df-func 17805 df-nat 17891 |
This theorem is referenced by: fuccocl 17914 fuclid 17916 fucrid 17917 fucass 17918 fucsect 17922 invfuc 17924 fucpropd 17927 evlfcllem 18171 evlfcl 18172 curfuncf 18188 yonedalem3a 18224 yonedalem3b 18229 yonedainv 18231 yonffthlem 18232 |
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