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Mirrors > Home > MPE Home > Th. List > offval3 | Structured version Visualization version 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 3459 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V) |
3 | elex 3459 | . . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
4 | 3 | adantl 485 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V) |
5 | dmexg 7594 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
6 | inex1g 5187 | . . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V) | |
7 | mptexg 6961 | . . . 4 ⊢ ((dom 𝐹 ∩ dom 𝐺) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
9 | 8 | adantr 484 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
10 | dmeq 5736 | . . . . 5 ⊢ (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹) | |
11 | dmeq 5736 | . . . . 5 ⊢ (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺) | |
12 | 10, 11 | ineqan12d 4141 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺)) |
13 | fveq1 6644 | . . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘𝑥) = (𝐹‘𝑥)) | |
14 | fveq1 6644 | . . . . 5 ⊢ (𝑏 = 𝐺 → (𝑏‘𝑥) = (𝐺‘𝑥)) | |
15 | 13, 14 | oveqan12d 7154 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → ((𝑎‘𝑥)𝑅(𝑏‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
16 | 12, 15 | mpteq12dv 5115 | . . 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 7389 | . . 3 ⊢ ∘f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥)))) | |
18 | 16, 17 | ovmpoga 7283 | . 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 1368 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 |
This theorem is referenced by: offres 7666 offsplitfpar 7798 ofco2 21056 dvsinax 42555 dvcosax 42568 fdivval 44953 |
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