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Mirrors > Home > MPE Home > Th. List > fnhomeqhomf | Structured version Visualization version GIF version |
Description: If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.) |
Ref | Expression |
---|---|
homffval.f | ⊢ 𝐹 = (Homf ‘𝐶) |
homffval.b | ⊢ 𝐵 = (Base‘𝐶) |
homffval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
fnhomeqhomf | ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnov 7284 | . 2 ⊢ (𝐻 Fn (𝐵 × 𝐵) ↔ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) | |
2 | homffval.f | . . . 4 ⊢ 𝐹 = (Homf ‘𝐶) | |
3 | homffval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | homffval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | 2, 3, 4 | homffval 16962 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
6 | eqeq2 2835 | . . 3 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻 ↔ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))) | |
7 | 5, 6 | mpbiri 260 | . 2 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻) |
8 | 1, 7 | sylbi 219 | 1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 × cxp 5555 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Basecbs 16485 Hom chom 16578 Homf chomf 16939 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-homf 16943 |
This theorem is referenced by: estrchomfeqhom 17388 rngchomfeqhom 44247 rngchomffvalALTV 44273 ringchomfeqhom 44293 |
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