Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > homffval | Structured version Visualization version GIF version |
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
Ref | Expression |
---|---|
homffval.f | ⊢ 𝐹 = (Homf ‘𝐶) |
homffval.b | ⊢ 𝐵 = (Base‘𝐶) |
homffval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
homffval | ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homffval.f | . 2 ⊢ 𝐹 = (Homf ‘𝐶) | |
2 | fveq2 6670 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | homffval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6670 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | homffval.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 7173 | . . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7229 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
10 | df-homf 16941 | . . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | |
11 | 3 | fvexi 6684 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 11, 11 | mpoex 7777 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V |
13 | 9, 10, 12 | fvmpt 6768 | . . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
14 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅) | |
15 | fvprc 6663 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
16 | 3, 15 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅) |
17 | 16 | olcd 870 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
18 | 0mpo0 7237 | . . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅) |
20 | 14, 19 | eqtr4d 2859 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
21 | 13, 20 | pm2.61i 184 | . 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
22 | 1, 21 | eqtri 2844 | 1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Basecbs 16483 Hom chom 16576 Homf chomf 16937 |
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-homf 16941 |
This theorem is referenced by: fnhomeqhomf 16961 homfval 16962 homffn 16963 homfeq 16964 oppchomf 16990 reschomf 17101 |
Copyright terms: Public domain | W3C validator |