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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 6839 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | homffval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6839 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | homffval.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 7368 | . . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7426 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
10 | df-homf 17510 | . . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | |
11 | 3 | fvexi 6853 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 11, 11 | mpoex 8004 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V |
13 | 9, 10, 12 | fvmpt 6945 | . . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
14 | fvprc 6831 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅) | |
15 | fvprc 6831 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
16 | 3, 15 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅) |
17 | 16 | olcd 872 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
18 | 0mpo0 7434 | . . . . 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 2780 | . . 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 2765 | 1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 ‘cfv 6493 (class class class)co 7351 ∈ cmpo 7353 Basecbs 17043 Hom chom 17104 Homf chomf 17506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-homf 17510 |
This theorem is referenced by: fnhomeqhomf 17531 homfval 17532 homffn 17533 homfeq 17534 oppchomf 17562 reschomf 17675 |
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