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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.) |
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 6103 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | homffval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6103 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | homffval.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 6566 | . . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦)) |
9 | 4, 4, 8 | mpt2eq123dv 6615 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
10 | df-homf 16154 | . . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | |
11 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
12 | 3, 11 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
13 | 12, 12 | mpt2ex 7136 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V |
14 | 9, 10, 13 | fvmpt 6191 | . . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
15 | mpt20 6623 | . . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)) = ∅ | |
16 | 15 | eqcomi 2619 | . . . 4 ⊢ ∅ = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)) |
17 | fvprc 6097 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅) | |
18 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
19 | 3, 18 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅) |
20 | mpt2eq12 6613 | . . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦))) | |
21 | 19, 19, 20 | syl2anc 691 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦))) |
22 | 16, 17, 21 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
23 | 14, 22 | pm2.61i 175 | . 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
24 | 1, 23 | eqtri 2632 | 1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 Hom chom 15779 Homf chomf 16150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-homf 16154 |
This theorem is referenced by: fnhomeqhomf 16174 homfval 16175 homffn 16176 homfeq 16177 oppchomf 16203 reschomf 16314 |
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