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| Mirrors > Home > MPE Home > Th. List > fsetfcdm | Structured version Visualization version GIF version | ||
| Description: The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| fsetfocdm.f | ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} | 
| fsetfocdm.s | ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) | 
| Ref | Expression | 
|---|---|
| fsetfcdm | ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3483 | . . . . 5 ⊢ 𝑔 ∈ V | |
| 2 | feq1 6715 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵)) | |
| 3 | fsetfocdm.f | . . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} | |
| 4 | 1, 2, 3 | elab2 3681 | . . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵) | 
| 5 | ffvelcdm 7100 | . . . . 5 ⊢ ((𝑔:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑔‘𝑋) ∈ 𝐵) | |
| 6 | 5 | expcom 413 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑔:𝐴⟶𝐵 → (𝑔‘𝑋) ∈ 𝐵)) | 
| 7 | 4, 6 | biimtrid 242 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑔 ∈ 𝐹 → (𝑔‘𝑋) ∈ 𝐵)) | 
| 8 | 7 | imp 406 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑔 ∈ 𝐹) → (𝑔‘𝑋) ∈ 𝐵) | 
| 9 | fsetfocdm.s | . 2 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) | |
| 10 | 8, 9 | fmptd 7133 | 1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {cab 2713 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 | 
| This theorem is referenced by: fsetfocdm 8902 | 
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