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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 7487 | . 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 17570 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) |
6 | eqeq2 2748 | . . 3 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻 ↔ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))) | |
7 | 5, 6 | mpbiri 257 | . 2 ⊢ (𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 × cxp 5631 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Basecbs 17083 Hom chom 17144 Homf chomf 17546 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-homf 17550 |
This theorem is referenced by: estrchomfeqhom 18023 rngchomfeqhom 46257 rngchomffvalALTV 46283 ringchomfeqhom 46303 |
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