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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcf | Structured version Visualization version GIF version |
Description: The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
ofcf.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) |
ofcf.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
ofcf.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcf.5 | ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
Ref | Expression |
---|---|
ofcf | ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffnd 6709 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | ofcf.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | ofcf.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
5 | eqidd 2725 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
6 | 2, 3, 4, 5 | ofcfval 33588 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶))) |
7 | 1 | ffvelcdmda 7077 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) |
8 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑇) |
9 | ofcf.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
10 | 9 | ralrimivva 3192 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
12 | ovrspc2v 7428 | . . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈) | |
13 | 7, 8, 11, 12 | syl21anc 835 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈) |
14 | 6, 13 | fmpt3d 7108 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∀wral 3053 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ∘f/c cofc 33585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-ofc 33586 |
This theorem is referenced by: signshf 34091 |
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