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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 17911 | . 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
6 | natcl.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | fveq2 6882 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
8 | fveq2 6882 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘𝑥) = (𝐾‘𝑋)) | |
9 | 7, 8 | oveq12d 7420 | . . 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 583 | 1 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4627 ‘cfv 6534 (class class class)co 7402 Xcixp 8888 Basecbs 17149 Hom chom 17213 Nat cnat 17900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-ixp 8889 df-func 17813 df-nat 17902 |
This theorem is referenced by: fuccocl 17925 fuclid 17927 fucrid 17928 fucass 17929 fucsect 17933 invfuc 17935 fucpropd 17938 evlfcllem 18182 evlfcl 18183 curfuncf 18199 yonedalem3a 18235 yonedalem3b 18240 yonedainv 18242 yonffthlem 18243 |
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