Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > comffn | Structured version Visualization version GIF version |
Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffn.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffn.b | ⊢ 𝐵 = (Base‘𝐶) |
comffn.h | ⊢ 𝐻 = (Hom ‘𝐶) |
comffn.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffn.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffn.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
comffn | ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) | |
2 | ovex 7246 | . . 3 ⊢ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓) ∈ V | |
3 | 1, 2 | fnmpoi 7840 | . 2 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) |
4 | comfffn.o | . . . 4 ⊢ 𝑂 = (compf‘𝐶) | |
5 | comfffn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
6 | comffn.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | eqid 2737 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
8 | comffn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | comffn.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | comffn.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | 4, 5, 6, 7, 8, 9, 10 | comffval 17202 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓))) |
12 | 11 | fneq1d 6472 | . 2 ⊢ (𝜑 → ((〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))) |
13 | 3, 12 | mpbiri 261 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 〈cop 4547 × cxp 5549 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 Basecbs 16760 Hom chom 16813 compcco 16814 compfccomf 17170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-comf 17174 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |