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Mirrors > Home > MPE Home > Th. List > natffn | Structured version Visualization version GIF version |
Description: The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
natrcl.1 | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
Ref | Expression |
---|---|
natffn | ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | natrcl.1 | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
2 | eqid 2727 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2727 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2727 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
5 | eqid 2727 | . . 3 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
6 | 1, 2, 3, 4, 5 | natfval 17929 | . 2 ⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(〈(𝑟‘𝑥), (𝑟‘𝑦)〉(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(〈(𝑟‘𝑥), (𝑠‘𝑥)〉(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))}) |
7 | ovex 7447 | . . . . . . 7 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V | |
8 | 7 | rgenw 3060 | . . . . . 6 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V |
9 | ixpexg 8934 | . . . . . 6 ⊢ (∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V → X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V |
11 | 10 | rabex 5328 | . . . 4 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(〈(𝑟‘𝑥), (𝑟‘𝑦)〉(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(〈(𝑟‘𝑥), (𝑠‘𝑥)〉(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ V |
12 | 11 | csbex 5305 | . . 3 ⊢ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(〈(𝑟‘𝑥), (𝑟‘𝑦)〉(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(〈(𝑟‘𝑥), (𝑠‘𝑥)〉(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ V |
13 | 12 | csbex 5305 | . 2 ⊢ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(〈(𝑟‘𝑥), (𝑟‘𝑦)〉(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(〈(𝑟‘𝑥), (𝑠‘𝑥)〉(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ V |
14 | 6, 13 | fnmpoi 8068 | 1 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∀wral 3056 {crab 3427 Vcvv 3469 ⦋csb 3889 〈cop 4630 × cxp 5670 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 1st c1st 7985 2nd c2nd 7986 Xcixp 8909 Basecbs 17173 Hom chom 17237 compcco 17238 Func cfunc 17833 Nat cnat 17924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-ixp 8910 df-func 17837 df-nat 17926 |
This theorem is referenced by: fuchom 17945 fuchomOLD 17946 |
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