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Mirrors > Home > MPE Home > Th. List > natfn | Structured version Visualization version GIF version |
Description: A natural transformation is a function on the objects of 𝐶. (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) |
Ref | Expression |
---|---|
natfn | ⊢ (𝜑 → 𝐴 Fn 𝐵) |
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 | eqid 2733 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
5 | 1, 2, 3, 4 | natixp 17696 | . 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥))) |
6 | ixpfn 8711 | . 2 ⊢ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)) → 𝐴 Fn 𝐵) | |
7 | 5, 6 | syl 17 | 1 ⊢ (𝜑 → 𝐴 Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 〈cop 4570 Fn wfn 6442 ‘cfv 6447 (class class class)co 7295 Xcixp 8705 Basecbs 16940 Hom chom 17001 Nat cnat 17685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-ixp 8706 df-func 17601 df-nat 17687 |
This theorem is referenced by: fuclid 17712 fucrid 17713 curfuncf 17984 yonedainv 18027 yonffthlem 18028 |
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