Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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/𝑐𝑅𝐶):𝐴⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcf.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | ffn 5958 | . . . 4 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | ofcf.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | ofcf.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
6 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
7 | 3, 4, 5, 6 | ofcfval 29487 | . 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑧 ∈ 𝐴 ↦ ((𝐹‘𝑧)𝑅𝐶))) |
8 | 1 | ffvelrnda 6267 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) |
9 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑇) |
10 | ofcf.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
11 | 10 | ralrimivva 2954 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
13 | ovrspc2v 6571 | . . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈) | |
14 | 8, 9, 12, 13 | syl21anc 1317 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧)𝑅𝐶) ∈ 𝑈) |
15 | 7, 14 | fmpt3d 6293 | 1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶):𝐴⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘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: signshf 29991 |
Copyright terms: Public domain | W3C validator |