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 29485 | . . . 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 790 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹) | |
4 | 3 | dmeqd 5248 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹) |
5 | 3 | fveq1d 6105 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥)) |
6 | simprr 792 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶) | |
7 | 5, 6 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶)) |
8 | 4, 7 | mpteq12dv 4663 | . . 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 6386 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
12 | 9, 10, 11 | syl2anc 691 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | ofcfval.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
14 | elex 3185 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
16 | fndm 5904 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
17 | 9, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
18 | 17, 10 | eqeltrd 2688 | . . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉) |
19 | mptexg 6389 | . . . 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 6686 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
22 | 17 | eleq2d 2673 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
23 | 22 | pm5.32i 667 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) |
24 | ofcfval.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
25 | 23, 24 | sylbi 206 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵) |
26 | 25 | oveq1d 6564 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶)) |
27 | 17, 26 | mpteq12dva 4662 | . 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
28 | 21, 27 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ∘𝑓/𝑐cofc 29484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ofc 29485 |
This theorem is referenced by: ofcval 29488 ofcfn 29489 ofcfeqd2 29490 ofcf 29492 ofcfval2 29493 ofcc 29495 ofcof 29496 |
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