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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 3149 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
3 | eqid 2798 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | fnmpt 6460 | . . . 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 6416 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)) |
8 | 5, 7 | mpbird 260 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
9 | fmpt2d.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐶) | |
10 | 9 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐶) |
11 | ffnfv 6859 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐶)) | |
12 | 8, 10, 11 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 |
This theorem is referenced by: cantnff 9121 limsupgre 14830 idaf 17315 curfcl 17474 mat2pmatf 21333 m2cpmf 21347 pm2mpf 21403 clsf 21653 kgenf 22146 lgamf 25627 vmaf 25704 lgsdchr 25939 mirf 26454 suppovss 30443 omsf 31664 erdszelem6 32556 cdleme50f 37838 dochfN 38652 binomcxplemdvsum 41059 |
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