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Mirrors > Home > ILE Home > Th. List > ofmres | GIF version |
Description: Equivalent expressions for a restriction of the function operation map. Unlike ∘𝑓 𝑅 which is a proper class, ( ∘𝑓 𝑅 ∣ ‘(A × B)) can be a set by ofmresex 5706, allowing it to be used as a function or structure argument. By ofmresval 5665, the restricted operation map values are the same as the original values, allowing theorems for ∘𝑓 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.) |
Ref | Expression |
---|---|
ofmres | ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 2959 | . . 3 ⊢ A ⊆ V | |
2 | ssv 2959 | . . 3 ⊢ B ⊆ V | |
3 | resmpt2 5541 | . . 3 ⊢ ((A ⊆ V ∧ B ⊆ V) → ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))) | |
4 | 1, 2, 3 | mp2an 402 | . 2 ⊢ ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) |
5 | df-of 5654 | . . 3 ⊢ ∘𝑓 𝑅 = (f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) | |
6 | 5 | reseq1i 4551 | . 2 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B)) |
7 | eqid 2037 | . . 3 ⊢ A = A | |
8 | eqid 2037 | . . 3 ⊢ B = B | |
9 | vex 2554 | . . . 4 ⊢ f ∈ V | |
10 | vex 2554 | . . . 4 ⊢ g ∈ V | |
11 | 9 | dmex 4541 | . . . . . 6 ⊢ dom f ∈ V |
12 | 11 | inex1 3882 | . . . . 5 ⊢ (dom f ∩ dom g) ∈ V |
13 | 12 | mptex 5330 | . . . 4 ⊢ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) ∈ V |
14 | 5 | ovmpt4g 5565 | . . . 4 ⊢ ((f ∈ V ∧ g ∈ V ∧ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) ∈ V) → (f ∘𝑓 𝑅g) = (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) |
15 | 9, 10, 13, 14 | mp3an 1231 | . . 3 ⊢ (f ∘𝑓 𝑅g) = (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) |
16 | 7, 8, 15 | mpt2eq123i 5510 | . 2 ⊢ (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) |
17 | 4, 6, 16 | 3eqtr4i 2067 | 1 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 Vcvv 2551 ∩ cin 2910 ⊆ wss 2911 ↦ cmpt 3809 × cxp 4286 dom cdm 4288 ↾ cres 4290 ‘cfv 4845 (class class class)co 5455 ↦ cmpt2 5457 ∘𝑓 cof 5652 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-of 5654 |
This theorem is referenced by: (None) |
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