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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 5764, allowing it to be used as a function or structure argument. By ofmresval 5723, 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 2965 | . . 3 ⊢ 𝐴 ⊆ V | |
2 | ssv 2965 | . . 3 ⊢ 𝐵 ⊆ V | |
3 | resmpt2 5599 | . . 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 5712 | . . 3 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | 5 | reseq1i 4608 | . 2 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) ↾ (𝐴 × 𝐵)) |
7 | eqid 2040 | . . 3 ⊢ 𝐴 = 𝐴 | |
8 | eqid 2040 | . . 3 ⊢ 𝐵 = 𝐵 | |
9 | vex 2560 | . . . 4 ⊢ 𝑓 ∈ V | |
10 | vex 2560 | . . . 4 ⊢ 𝑔 ∈ V | |
11 | 9 | dmex 4598 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
12 | 11 | inex1 3891 | . . . . 5 ⊢ (dom 𝑓 ∩ dom 𝑔) ∈ V |
13 | 12 | mptex 5387 | . . . 4 ⊢ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) ∈ V |
14 | 5 | ovmpt4g 5623 | . . . 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 1232 | . . 3 ⊢ (𝑓 ∘𝑓 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) |
16 | 7, 8, 15 | mpt2eq123i 5568 | . 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 2070 | 1 ⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∩ cin 2916 ⊆ wss 2917 ↦ cmpt 3818 × cxp 4343 dom cdm 4345 ↾ cres 4347 ‘cfv 4902 (class class class)co 5512 ↦ cmpt2 5514 ∘𝑓 cof 5710 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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