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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcfval | Structured version Visualization version GIF version | ||
| Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
| Ref | Expression |
|---|---|
| ofcfval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| ofcfval.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| ofcfval.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
| ofcfval.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
| Ref | Expression |
|---|---|
| ofcfval | ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ofc 34097 | . . . 4 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))) |
| 3 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹) | |
| 4 | 3 | dmeqd 5916 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹) |
| 5 | 3 | fveq1d 6908 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥)) |
| 6 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶) | |
| 7 | 5, 6 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶)) |
| 8 | 4, 7 | mpteq12dv 5233 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
| 9 | ofcfval.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 10 | ofcfval.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 11 | fnex 7237 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
| 12 | 9, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
| 13 | ofcfval.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
| 14 | 13 | elexd 3504 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
| 15 | 9 | fndmd 6673 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 16 | 15, 10 | eqeltrd 2841 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉) |
| 17 | 16 | mptexd 7244 | . . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) |
| 18 | 2, 8, 12, 14, 17 | ovmpod 7585 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
| 19 | 15 | eleq2d 2827 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
| 20 | 19 | pm5.32i 574 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) |
| 21 | ofcfval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
| 22 | 20, 21 | sylbi 217 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵) |
| 23 | 22 | oveq1d 7446 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶)) |
| 24 | 15, 23 | mpteq12dva 5231 | . 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
| 25 | 18, 24 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ∘f/c cofc 34096 |
| 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 |
| 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-ofc 34097 |
| This theorem is referenced by: ofcval 34100 ofcfn 34101 ofcfeqd2 34102 ofcf 34104 ofcfval2 34105 ofcc 34107 ofcof 34108 |
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