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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 3501 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V) |
| 3 | elex 3501 | . . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) | |
| 4 | 3 | adantl 481 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V) |
| 5 | dmexg 7923 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
| 6 | inex1g 5319 | . . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V) | |
| 7 | mptexg 7241 | . . . 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 480 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
| 10 | dmeq 5914 | . . . . 5 ⊢ (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹) | |
| 11 | dmeq 5914 | . . . . 5 ⊢ (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺) | |
| 12 | 10, 11 | ineqan12d 4222 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺)) |
| 13 | fveq1 6905 | . . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘𝑥) = (𝐹‘𝑥)) | |
| 14 | fveq1 6905 | . . . . 5 ⊢ (𝑏 = 𝐺 → (𝑏‘𝑥) = (𝐺‘𝑥)) | |
| 15 | 13, 14 | oveqan12d 7450 | . . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → ((𝑎‘𝑥)𝑅(𝑏‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
| 16 | 12, 15 | mpteq12dv 5233 | . . 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 7697 | . . 3 ⊢ ∘f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥)))) | |
| 18 | 16, 17 | ovmpoga 7587 | . 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 1373 | 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
| This theorem is referenced by: offres 8008 offsplitfpar 8144 ofco2 22457 dvsinax 45928 dvcosax 45941 fdivval 48460 |
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