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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 2824 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2824 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | comfffval 16971 | . 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 7192 | . . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V | |
7 | fvex 6686 | . . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V | |
8 | 6, 7 | mpoex 7780 | . 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V |
9 | 5, 8 | fnmpoi 7771 | 1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 × cxp 5556 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 2nd c2nd 7691 Basecbs 16486 Hom chom 16579 compcco 16580 compfccomf 16941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-comf 16945 |
This theorem is referenced by: (None) |
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