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Mathbox for Thierry Arnoux |
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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 31465 | . . . 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 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹) | |
4 | 3 | dmeqd 5738 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹) |
5 | 3 | fveq1d 6647 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥)) |
6 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶) | |
7 | 5, 6 | oveq12d 7153 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶)) |
8 | 4, 7 | mpteq12dv 5115 | . . 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 6957 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
12 | 9, 10, 11 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | ofcfval.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
14 | 13 | elexd 3461 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
15 | 9 | fndmd 6427 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
16 | 15, 10 | eqeltrd 2890 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉) |
17 | 16 | mptexd 6964 | . . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) |
18 | 2, 8, 12, 14, 17 | ovmpod 7281 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
19 | 15 | eleq2d 2875 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
20 | 19 | pm5.32i 578 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) |
21 | ofcfval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
22 | 20, 21 | sylbi 220 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵) |
23 | 22 | oveq1d 7150 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶)) |
24 | 15, 23 | mpteq12dva 5114 | . 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
25 | 18, 24 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ↦ cmpt 5110 dom cdm 5519 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ∘f/c cofc 31464 |
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 |
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-ofc 31465 |
This theorem is referenced by: ofcval 31468 ofcfn 31469 ofcfeqd2 31470 ofcf 31472 ofcfval2 31473 ofcc 31475 ofcof 31476 |
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