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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 3482 | . . . . 5 ⊢ 𝑔 ∈ V | |
2 | feq1 6717 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵)) | |
3 | fsetfocdm.f | . . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} | |
4 | 1, 2, 3 | elab2 3685 | . . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵) |
5 | ffvelcdm 7101 | . . . . 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 7134 | 1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: fsetfocdm 8900 |
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