Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17224 | . 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
6 | natcl.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | fveq2 6672 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
8 | fveq2 6672 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐾‘𝑥) = (𝐾‘𝑋)) | |
9 | 7, 8 | oveq12d 7176 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) = ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
10 | 9 | fvixp 8468 | . 2 ⊢ ((𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ 𝑋 ∈ 𝐵) → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
11 | 5, 6, 10 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 Basecbs 16485 Hom chom 16578 Nat cnat 17213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-ixp 8464 df-func 17130 df-nat 17215 |
This theorem is referenced by: fuccocl 17236 fuclid 17238 fucrid 17239 fucass 17240 fucsect 17244 invfuc 17246 fucpropd 17249 evlfcllem 17473 evlfcl 17474 curfuncf 17490 yonedalem3a 17526 yonedalem3b 17531 yonedainv 17533 yonffthlem 17534 |
Copyright terms: Public domain | W3C validator |