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Mirrors > Home > ILE Home > Th. List > offval3 | GIF version |
Description: General value of (𝐹 ∘𝑓 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
offval3 | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2737 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V) |
3 | elex 2737 | . . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
4 | 3 | adantl 275 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V) |
5 | dmexg 4868 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
6 | inex1g 4118 | . . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V) | |
7 | mptexg 5710 | . . . 4 ⊢ ((dom 𝐹 ∩ dom 𝐺) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) | |
8 | 5, 6, 7 | 3syl 17 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
9 | 8 | adantr 274 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
10 | dmeq 4804 | . . . . 5 ⊢ (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹) | |
11 | dmeq 4804 | . . . . 5 ⊢ (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺) | |
12 | 10, 11 | ineqan12d 3325 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺)) |
13 | fveq1 5485 | . . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘𝑥) = (𝐹‘𝑥)) | |
14 | fveq1 5485 | . . . . 5 ⊢ (𝑏 = 𝐺 → (𝑏‘𝑥) = (𝐺‘𝑥)) | |
15 | 13, 14 | oveqan12d 5861 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → ((𝑎‘𝑥)𝑅(𝑏‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
16 | 12, 15 | mpteq12dv 4064 | . . 3 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
17 | df-of 6050 | . . 3 ⊢ ∘𝑓 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥)))) | |
18 | 16, 17 | ovmpoga 5971 | . 2 ⊢ ((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)))) |
19 | 2, 4, 9, 18 | syl3anc 1228 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1343 ∈ wcel 2136 Vcvv 2726 ∩ cin 3115 ↦ cmpt 4043 dom cdm 4604 ‘cfv 5188 (class class class)co 5842 ∘𝑓 cof 6048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 |
This theorem is referenced by: offres 6103 |
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