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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 6885 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | homffval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | eqtr4di 2784 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6885 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | homffval.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 7422 | . . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7480 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
10 | df-homf 17623 | . . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | |
11 | 3 | fvexi 6899 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 11, 11 | mpoex 8065 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V |
13 | 9, 10, 12 | fvmpt 6992 | . . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
14 | fvprc 6877 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅) | |
15 | fvprc 6877 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
16 | 3, 15 | eqtrid 2778 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅) |
17 | 16 | olcd 871 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
18 | 0mpo0 7488 | . . . . 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 2769 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
21 | 13, 20 | pm2.61i 182 | . 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
22 | 1, 21 | eqtri 2754 | 1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∅c0 4317 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17153 Hom chom 17217 Homf chomf 17619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-homf 17623 |
This theorem is referenced by: fnhomeqhomf 17644 homfval 17645 homffn 17646 homfeq 17647 oppchomf 17675 reschomf 17788 |
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