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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 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | comfffval 16968 | . 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 7189 | . . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V | |
7 | fvex 6683 | . . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V | |
8 | 6, 7 | mpoex 7777 | . 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V |
9 | 5, 8 | fnmpoi 7768 | 1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 × cxp 5553 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 compcco 16577 compfccomf 16938 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-comf 16942 |
This theorem is referenced by: (None) |
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