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Mirrors > Home > MPE Home > Th. List > fmpt2d | Structured version Visualization version GIF version |
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.) |
Ref | Expression |
---|---|
fmpt2d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
fmpt2d.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
fmpt2d.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐶) |
Ref | Expression |
---|---|
fmpt2d | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpt2d.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
2 | 1 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
3 | eqid 2732 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | fnmpt 6687 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
6 | fmpt2d.1 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
7 | 6 | fneq1d 6639 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)) |
8 | 5, 7 | mpbird 256 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
9 | fmpt2d.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐶) | |
10 | 9 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐶) |
11 | ffnfv 7114 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐶)) | |
12 | 8, 10, 11 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ↦ cmpt 5230 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 |
This theorem is referenced by: cantnff 9665 limsupgre 15421 idaf 18009 curfcl 18181 mat2pmatf 22221 m2cpmf 22235 pm2mpf 22291 clsf 22543 kgenf 23036 lgamf 26535 vmaf 26612 lgsdchr 26847 mirf 27900 suppovss 31893 ghmquskerlem3 32519 ghmqusker 32520 omsf 33283 erdszelem6 34175 cdleme50f 39401 dochfN 40215 binomcxplemdvsum 43099 |
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