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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/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ofc 30764 | . . . 4 ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))) |
3 | simprl 761 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹) | |
4 | 3 | dmeqd 5573 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹) |
5 | 3 | fveq1d 6450 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥)) |
6 | simprr 763 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶) | |
7 | 5, 6 | oveq12d 6942 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶)) |
8 | 4, 7 | mpteq12dv 4971 | . . 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 6755 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
12 | 9, 10, 11 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | ofcfval.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
14 | elex 3414 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
16 | fndm 6237 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
17 | 9, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
18 | 17, 10 | eqeltrd 2859 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉) |
19 | mptexg 6758 | . . . 4 ⊢ (dom 𝐹 ∈ 𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) |
21 | 2, 8, 12, 15, 20 | ovmpt2d 7067 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
22 | 17 | eleq2d 2845 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
23 | 22 | pm5.32i 570 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) |
24 | ofcfval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
25 | 23, 24 | sylbi 209 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵) |
26 | 25 | oveq1d 6939 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶)) |
27 | 17, 26 | mpteq12dva 4970 | . 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
28 | 21, 27 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ↦ cmpt 4967 dom cdm 5357 Fn wfn 6132 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 ∘𝑓/𝑐cofc 30763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-ofc 30764 |
This theorem is referenced by: ofcval 30767 ofcfn 30768 ofcfeqd2 30769 ofcf 30771 ofcfval2 30772 ofcc 30774 ofcof 30775 |
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