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Mirrors > Home > MPE Home > Th. List > comffval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval2.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffval2.b | ⊢ 𝐵 = (Base‘𝐶) |
comfffval2.h | ⊢ 𝐻 = (Homf ‘𝐶) |
comfffval2.x | ⊢ · = (comp‘𝐶) |
comffval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffval2.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
comffval2 | ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval2.o | . . 3 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffval2.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | comfffval2.x | . . 3 ⊢ · = (comp‘𝐶) | |
5 | comffval2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | comffval2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | comffval2.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | comffval 16952 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
9 | comfffval2.h | . . . 4 ⊢ 𝐻 = (Homf ‘𝐶) | |
10 | 9, 2, 3, 6, 7 | homfval 16945 | . . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍)) |
11 | 9, 2, 3, 5, 6 | homfval 16945 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌)) |
12 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓) = (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) | |
13 | 10, 11, 12 | mpoeq123dv 7215 | . 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
14 | 8, 13 | eqtr4d 2859 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4559 ‘cfv 6341 (class class class)co 7142 ∈ cmpo 7144 Basecbs 16466 Hom chom 16559 compcco 16560 Homf chomf 16920 compfccomf 16921 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-homf 16924 df-comf 16925 |
This theorem is referenced by: (None) |
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