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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 5922, allowing it to be used as a function or structure argument. By ofmresval 5881, 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 3047 | . . 3 ⊢ 𝐴 ⊆ V | |
2 | ssv 3047 | . . 3 ⊢ 𝐵 ⊆ V | |
3 | resmpt2 5757 | . . 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 418 | . 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 5870 | . . 3 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | 5 | reseq1i 4722 | . 2 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵)) |
7 | eqid 2089 | . . 3 ⊢ 𝐴 = 𝐴 | |
8 | eqid 2089 | . . 3 ⊢ 𝐵 = 𝐵 | |
9 | vex 2623 | . . . 4 ⊢ 𝑓 ∈ V | |
10 | vex 2623 | . . . 4 ⊢ 𝑔 ∈ V | |
11 | 9 | dmex 4712 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
12 | 11 | inex1 3979 | . . . . 5 ⊢ (dom 𝑓 ∩ dom 𝑔) ∈ V |
13 | 12 | mptex 5537 | . . . 4 ⊢ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V |
14 | 5 | ovmpt4g 5781 | . . . 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 1274 | . . 3 ⊢ (𝑓 ∘𝑓 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) |
16 | 7, 8, 15 | mpt2eq123i 5726 | . 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 2119 | 1 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∈ wcel 1439 Vcvv 2620 ∩ cin 2999 ⊆ wss 3000 ↦ cmpt 3905 × cxp 4450 dom cdm 4452 ↾ cres 4454 ‘cfv 5028 (class class class)co 5666 ↦ cmpt2 5668 ∘𝑓 cof 5868 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-of 5870 |
This theorem is referenced by: (None) |
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