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Mirrors > Home > MPE Home > Th. List > comfffn | 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) |
Ref | Expression |
---|---|
comfffn | ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffn.o | . . 3 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffn.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | eqid 2622 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2622 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | comfffval 16358 | . 2 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓))) |
6 | ovex 6678 | . . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V | |
7 | fvex 6201 | . . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V | |
8 | 6, 7 | mpt2ex 7247 | . 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V |
9 | 5, 8 | fnmpt2i 7239 | 1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 2nd c2nd 7167 Basecbs 15857 Hom chom 15952 compcco 15953 compfccomf 16328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-comf 16332 |
This theorem is referenced by: (None) |
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